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WHEN  I  showed  you  the  last  sheet  of  my  History  of  the  In 
ductive  Sciences  in  its  transit  through  the  press,  you  told  me  that 
I  ought  to  add  a  paragraph  or  two  at  the  end,  by  way  of  Moral 
to  the  story ;  and  I  replied  that  the  Moral  would  be  as  long  as 
the  story  itself.  The  present  work,  the  Moral  which  you  then 
desired,  I  have,  with  some  effort,  reduced  within  a  somewhat 
smaller  compass  than  I  then  spoke  of;  and  I  cannot  dedicate  it 
to  any  one  with  so  much  pleasure  as  to  you. 

It  has  always  been  my  wish  that,  as  far  and  as  long  as  men 
might  know  anything  of  me  by  my  writings,  they  should  hear  of  me 
along  with  the  friends  with  whom  I  have  lived,  whom  I  have  loved, 
and  by  whose  conversation  I  have  been  animated  to  hope  that  I 
too  might  add  something  to  the  literature  of  our  country.  There 
is  no  one  whose  name  has,  on  such  grounds,  a  better  claim  than 
yours  to  stand  in  the  front  of  a  work,  which  has  been  the  subject 
of  my  labours  for  no  small  portion  of  our  long  period  of  friend 
ship.  But  there  is  another  reason  which  gives  a  peculiar  pro 
priety  to  this  dedication  of  my  Philosophy  to  you.  I  have  littlo 
doubt  that  if  your  life  had  not  been  absorbed  in  struggling 
with  many  of  the  most  difficult  problems  of  a  difficult  science, 
you  would  have  been  my  fellow-labourer  or  master  in  the  work 
which  I  have  here  undertaken.  The  same  spirit  which  dictated 
your  vigorous  protest  against  some  of  the  errours  which  I  also 
attempt  to  expose,  would  have  led  you,  if  your  thoughts  had  been 



more  free,  to  take  a  leading  share  in  that  Reform  of  Philosophy, 
which  all  who  are  alive  to  such  errours,  must  see  to  be  now  in 
dispensable.  To  you  I  may  most  justly  inscribe  a. work  which 
contains  a  criticism  of  the  fallacies  of  the  ultra-Lockian  school. 

I  will  mention  one  other  reason  which  enters  into  the  satisfac 
tion  with  which  I  place  your  name  at  the  head  of  my  Philosophy. 
By  doing  so,  I  may  consider  myself  as  dedicating  it  to  the  College 
to  which  we  both  belong,  to  which  we  both  owe  so  much  of  all 
that  we  arc,  and  in  which  we  have  lived  together  so  long  and  so 
happily;  and  that,  be  it  remembered,  the  College  of  Bacon  and  of 
Newton.  That  College,  I  know,  holds  a  strong  place  in  your  affec 
tions,  as  in  mine ;  and  among  many  reasons,  not  least  on  this 
account ; — we  believe  that  sound  and  enduring  philosophy  ever 
finds  there  a  congenial  soil  and  a  fostering  shelter.  If  the  doc 
trines  which  the  present  work  contains  be  really  true  and  valu 
able,  my  unhesitating  trust  is,  that  they  will  spread  gradually 
from  these  precincts  to  every  part  of  the  land. 

That  this  office  of  being  the  fosterer  and  diffuser  of  truth  may 
ever  belong  to  our  common  Nursing  Mother,  and  that  you,  my 
dear  Sedgwick,  may  long  witness  and  contribute  to  these  bene 
ficial  influences,  is  the  hearty  wish  of 

Yours  affectionately, 


Trinity  College,  May  ^,  1840. 




1\  the  Preface  to  the  first  edition  of  this  work,  it  was 
stated  that  the  work  was  intended  as  an  application  of 
the  plan  of  Bacon's  Novum  Oryanon  to  the  present  conr 
dition  of  Physical  Science.  Such  an  undertaking,  it  was 
there  said,  plainly  belongs  to  the  present  generation. 
Bacon  only  divined  how  sciences  might  be  constructed ; 
we  can  trace,  in  their  history,  how  their  construction 
has  taken  place.  However  sagacious  were  his  conjec 
tures,  it  may  be  expected  that  they  will  be  further  illus 
trated  by  facts  which  we  know  to  have  really  occurred. 
However  large  were  his  anticipations,  the  actual  progress 
of  science  since  his  time  may  aid  in  giving  comprehen 
siveness  to  our  views.  And  with  respect  to  the  methods 
by  which  science  is  to  be  promoted, — the  structure  and 
operation  of  the  Oryan  by  which  truth  is  to  be  collected 
from  nature, — we  know  that,  though  Bacon's  general 
maxims  still  guide  and  animate  philosophical  enquirers 
yet  that  his  views,  in  their  detail,  have  all  turned  out 
inapplicable  :  the  technical  parts  of  his  method  failed  in 
his  hands,  and  are  forgotten  among  the  cultivators  of 
science.  It  cannot  be  an  unfit  task,  at  the  present  day, 
to  endeavour  to  extract  from  the  actual  past  progress 
of  science,  the  elements  of  a  more  effectual  and  sub- 


stantial  Method  of  Discovery.  The  advances  which 
have,  during  the  last  three  centuries,  been  made  in  the 
physical  sciences ; — in  Astronomy,  in  Physics,  in  Che 
mistry,  in  Natural  History,  in  Physiology; — these  are 
allowed  by  all  to  be  real,  to  be  great,  to  be  striking : 
may  it  not  be,  then,  that  these  steps  of  progress  have 
in  them  something  alike? — that  in  each  advancing  move 
ment  there  is  some  common  process,  some  common  prin 
ciple  ? — that  the  organ  by  which  discoveries  have  been 
made  has  had  something  uniform  in  its  structure  and 
working  ?  If  this  be  so,  and  if  we  can,  by  attending  to 
the  past  history  of  science,  discover  something  of  this 
common  element  and  common  process  in  all  discoveries, 
we  shall  have  a  Philosophy  of  Science,  such  as  our  times 
may  naturally  hope  for : — we  shall  have  the  New  Organ 
of  Bacon,  renovated  according  to  our  advanced  intellec 
tual  position  and  office. 

It  was  with  the  view  to  such  a  continuation  and 
extension  of  Bacon's  design,  that  I  undertook  that  sur 
vey  of  the  History  of  Science  which  I  have  given  in 
another  work ;  and  that  analysis  of  the  advance  of  each 
science  which  the  present  work  contains.  Of  the  doc 
trines  promulgated  by  Bacon,  none  has  more  completely 
remained  with  us,  as  a  stable  and  valuable  truth,  than 
his  declaration  that  true  knowledge  is  to  be  obtained 
from  Facts  by  Induction :  and  in  order  to  denote  that  I 
start  at  once  from  the  point  to  which  Bacon  thus  led  us, 
I  have,  both  in  the  History  and  in  the  Philosophy,  termed 
the  sciences  with  which  I  have  to  do,  the  Inductive  Sci 
ences.  By  treating  of  the  Physical  Sciences  only,  while 
I  speak  of  the  Inductive  Sciences  in  the  description  of 


ray  design,  I  do  not,  (as  I  have  already  elsewhere  said*) 
intend  to  deny  the  character  of  Inductive  Sciences  to 
many  other  branches  of  knowledge,  as  for  instance,  Eth 
nology,  Glossology,  Political  Economy,  and  Psychology. 
But  I  think  it  will  be  allowed  that  by  taking,  as  I  have 
done,  the  Physical  Sciences  alone,  in  which  the  truths 
established  are  universally  assented  to,  and  regarded  with 
comparative  calmness,  we  are  better  able  to  discuss  the 
formal  conditions  and  general  processes  of  scientific 
discovery,  than  we  could  do  if  we  entangled  ourselves 
among  subjects  where  the  interest  is  keener  and  the 
truth  more  controverted.  Perhaps  a  more  exact  descrip 
tion  of  the  present  work  would  be,  The  Philosophy  of 
the  Inductive  Sciences,  founded  upon  the  History  of  the 
principal  Physical  Sciences. 

I  am  well  aware  how  much  additional  interest  and 
attractiveness  are  given  to  speculations  concerning  the 
progress  of  human  knowledge,  when  we  include  in  them, 
as  examples  of  such  knowledge,  views  on  subjects  of 
politics,  morals,  beauty  in  art  and  literature,  and  the  like. 
Prominent  instances  of  the  effect  of  this  mode  of  treating 
such  subjects  have  recently  appeared.  But  I  still  think 
that  the  real  value  and  import  of  Inductive  Philosophy, 
even  in  its  application  to  such  subjects,  are  best  brought 
into  view  by  making  the  progress  of  political,  and  moral 
and  callesthetical^  truth  a  subject  of  consideration  apart 
from  physical  science. 

It  can  hardly  happen  that  a  work  which  treats  of 
Methods  of  Scientific  Discovery  shall  not  seem  to  fail  in 

*  Hist.  I  ml.  Sci.     Second  Edition.     Note  to  the  Introduction, 
t  Sec  Vol.  ii.     On  the  Language  of  Science,  Aphorism,  xvn. 


the  positive  results  which  it  offers.  For  an  Art  of  Dis 
covery  is  not  possible.  At  each  step  of  the  progress  of 
science,  are  needed  invention,  sagacity,  genius ; — elements 
which  no  Art  can  give.  We  may  hope  in  vain,  as  Bacon 
hoped,  for  an  organ  which  shall  enable  all  men  to  construct 
scientific  truths,  as  a  pair  of  compasses  enables  all  men 
to  construct  exact  circles  *.  The  practical  results  of  the 
Philosophy  of  Science  must,  we  are  persuaded,  be  rather 
classification  and  analysis  than  precept  and  method.  1 
think  however  that  the  methods  of  discovery  which 
I  have  to  recommend,  though  gathered  from  a  wider 
survey  of  scientific  history,  as  to  subject  and  as  to 
time,  than,  (so  far  as  I  am  aware,)  has  been  elsewhere 
attempted,  are  quite  as  definite  and  practical  as  any 
others  which  have  been  proposed ;  with  the  great  addi 
tional  advantage  of  being  the  methods  by  which  all  great 
discoveries  in  physical  science  really  have  been  made. 
This  may  be  said,  for  instance,  of  the  Method  of  Grada 
tion,  and  the  Method  of  Natural  Classification,  spoken 
of  Book  XIIL  Chap.  vm. ;  and  in  a  narrower  sense,  of 
the  Method  of  Curves,  the  Method  of  Means,  the  Method 
of  Least  Squares,  and  the  Method  of  Residues,  spoken 
of  in  Chap.  vn.  of  the  same  Book.  Also  the  Remarks 
on  the  Use  of  Hypotheses  and  on  the  Tests  of  Hypotheses 
(Book  xi.  Chap,  v.)  point  out  features  which  mark  the 

usual  course  of  discoverv. 


But  undoubtedly  one  of  the  principal  lessons  which 
results  from  the  views  here  given  is  that  different 
sciences  may  be  expected  to  advance  by  different  modes 
of  procedure,  according  to  their  present  condition ;  and 

*'   AV.  Or*,.  Lib.  i.  Apli.  (il 


that,  in  many  of  these  sciences,  an  Induction  per 
formed  by  any  of  the  methods  just  referred  to,  is  not 
the  step  which  we  may  expect  to  see  next  made. 
Several  of  the  sciences  may  not  be  in  a  condition  which 
fits  them  for  such  a  Colligation  of  Facts,  (to  use  the 
phraseology  to  which  the  succeeding  analysis  has  led 
me.  See  B.  xi.  C.  i).  The  Facts  may,  at  the  present 
time,  require  to  be  more  fully  observed,  or  the  Idea  by 
which  they  are  to  be  colligated  may  require  to  be  more 
fully  unfolded. 

But  in  this  point  also,  our  speculations  are  far  from 
being  barren  of  practical  results.  The  Philosophy  of 
each  Science,  as  given  in  the  present  work,  affords  us 
means  of  discerning  whether  that  which  is  needed  for 
the  further  progress  of  the  Science  has  its  place  in  the 
Observations,  or  in  the  Ideas,  or  in  the  union  of  the  two. 
If  Observations  be  wanted,  the  Methods  of  Observation 
given  in  Book  xm.  Chap.  u.  may  be  referred  to ;  if 
those  who  are  to  make  the  next  discoveries  need,  for 
that  purpose,  a  developement  of  their  Ideas,  the  modes 
in  which  such  a  developement  has  usually  taken  place 
are  treated  of  in  Chapters  in.  and  iv.  of  that  Book. 

Perhaps  one  of  the  most  prominent  points  of  this 
work  is  the  attempt  to  show  the  place  which  discussions 
concerning  Ideas  have  had  in  the  progress  of  science. 
The  metaphysical  aspect  of  each  of  the  physical  sciences 
is  very  far  from  being,  as  some  have  tried  to  teach,  an 
aspect  which  it  passes  through  previously  to  the  most 
derided  progress  of  the  science.  On  the  contrary,  the 
metaphysics]  is  a  necessary  part  of  the  inductive  move 
ment.  This,  which  is  evidently  so  by  the  nature  of  the 


case,  is  proved  by  a  copious  collection  of  historical  evi 
dences  in  the  first  ten  Books  of  the  present  work.  Those 
Books  contain  an  account  of  the  principal  philosophical 
controversies  which  have  taken  place  in  all  the  physical 
sciences,  from  Mathematics  to  Physiology ;  and  these 
controversies,  which  must  be  called  metaphysical  if  any 
thing  be  so  called,  have  been  conducted  by  the  greatest 
discoverers  in  each  science,  and  have  been  an  essential 
part  of  the  discoveries  made.  Physical  discoverers  have 
differed  from  barren  speculators,  not  by  having  no  meta 
physics  in  their  heads,  but  by  having  good  metaphysics 
while  their  adversaries  had  bad ;  and  by  binding  their 
metaphysics  to  their  physics,  instead  of  keeping  the  two 
asunder.  I  trust  that  the  ten  Books  of  which  I  have 
spoken  are  of  some  value,  even  as  a  series  of  analyses  of 
a  number  of  remarkable  controversies  ;  but  I  cannot  con 
ceive  how  any  one,  after  reading  these  Books,  can  fail 
to  see  that  there  is  in  progressive  science  a  metaphysical 
as  well  as  a  physical  element ; — ideas,  as  well  as  facts,— 
thoughts,  as  w^ell  as  things : — in  short,  that  the  Funda 
mental  Antithesis,  for  which  I  contend,  is  there  most 
abundantly  and  strikingly  exemplified. 

On  the  subject  of  this  doctrine  of  a  Fundamental 
Analysis,  which  our  knowledge  always  involves,  I  will 
venture  here  to  add  a  remark,  which  looks  beyond  the 
domain  of  the  physical  sciences.  This  doctrine  is  suited 
to  throw  light  upon  Moral  and  Political  Philosophy,  no 
less  than  upon  Physical.  In  Morality,  in  Legislation,  in 
National  Polity,  we  have  still  to  do  with  the  opposition 
and  combination  of  two  Elements  ; — of  Facts  and  Ideas ; 
of  Tlistorv,  and  an  Ideal  Standard  of  Action ;  of  actual 


character  and  position,  and  of  the  aims  which  are  placed 
above  the  Actual.  Each  of  these  is  in  conflict  with  the 
other ;  each  modifies  and  moulds  the  other.  We  can  never 
escape  the  control  of  the  first ;  we  must  .ever  cease  to 
strive  to  extend  the  sway  of  the  second.  In  these  cases, 
indeed,  the  Ideal  Element  assumes  a  new  form.  It  in 
cludes  the  Idea  of  Duty.  The  opposition,  the  action 
and  re-action,  the  harmony  at  which  we  must  ever 
aim,  and  can  never  reach,  are  between  what  is  and  what 
ought  to  be ; — between  the  past  or  present  Fact,  and 
the  Supreme  Idea.  The  Idea  can  never  be  independ 
ent  of  the  Fact,  but  the  Fact  must  ever  be  drawn 
towards  the  Idea.  The  History  of  Human  Societies, 
and  of  each  Individual,  is  by  the  moral  philosopher, 
regarded  in  reference  to  this  Antithesis ;  and  thus  both 
Public  and  Private  Morality  becomes  an  actual  progress 
towards  an  Ideal  Form ;  or  ceases  to  be  a  moral  reality. 

I  have  made  very  slight  alterations  in  the  first 
edition,  except  that  the  First  Book  is  remodelled  with 
a  view  of  bringing  out  more  clearly  the  basis  of  the 
work ; — this  doctrine  of  the  Fundamental  Antithesis  of 
Philosophy.  This  doctrine,  and  its  relation  to  the  rest 
of  the  work,  have  become  more  clear  in  the  years 
which  have  elapsed  since  the  first  edition. 

A  separate  Essay,  in  which  this  doctrine  was  ex 
plained,  and  a  few  other  Essays  previously  published  in 
various  forms,  and  containing  discussions  of  special 
points  belonging  to  the  scheme  of  philosophy  here  de 
livered,  have  attracted  some  notice,  both  in  this  and  in 
other  countries.  I  have  therefore  added  them  as  an 
Appendix  to  the  present  edition. 

Xll  PREFACi;    TO 

I  have  added  a  few  Notes,  in  answer  to  arguments 
brought  against  particular  parts  of  this  work.  I  have 
written  these  in  what  I  have  elsewhere  called  an  im 
personal  manner ;  wishing  to  avoid  controversy,  so  far 
as  justice  to  philosophical  Truth  will  allow  me  to  do  so. 

I  have  not  given  any  detailed  reply  to  the  criticisms 
of  this  work  which  occur  in  Mr.  Mill's  System  of  Logic. 
The  consideration  of  these  criticisms  would  be  interest 
ing  to  me,  and  I  think  would  still  further  establish  the 
doctrines  which  I  have  here  delivered.  But  such  a  dis 
cussion  would  involve  me  in  a  critique  of  Mr.  Mill's 
work ;  which  if  1  were  to  offer  to  the  world,  I  should 
think  it  more  suitable  to  publish  separately. 

More  than  one  of  my  critics  has  expressed  an  opinion 
that  when  I  published  this  work,  I  had  not  given  due  at 
tention  to  the  Cours  de  Philosophic  Positive  of  M.  Comte. 
I  had,  and  have,  an  opinion  of  the  value  of  M.  Comte's 
speculations  very  different  from  that  entertained  by  my 
monitors.  I  had  in  the  former  edition  discussed,  and, 
as  I  conceive,  confuted,  some  of  M.  Comte's  leading- 
doctrines*.  In  order  further  to  show  that  I  had  not 
lightly  passed  over  those  portions  of  M.  Comte's  work 
which  had  then  appeared,  I  now  publish  f  an  additional 
portion  of  a  critique  of  the  work  which,  though  I  had 
written,  I  excluded  from  the  former  edition.  This  is 
printed  exactly  as  it  existed  in  manuscript  at  the 
period  of  that  publication.  To  return  to  the  subject  and 
to  take  it  up  in  all  its  extent,  would  be  an  undertaking 
out  of  the  range  of  a  new  edition  of  my  published 

*    B.  xi.  r.  vii.      B.  \ni.  c.  iv.  I    I>.  \n.  c.  xvi. 


Bacon  delivered  his  philosophy  in  Aphorisms ; — a 
series  of  Sentences  which  profess  to  exhibit  rather  the 
results  of  thought  than  the  process  of  thinking.  A 
mere  Aphoristic  Philosophy  unsupported  by  reasoning, 
is  not  suited  to  the  present  time.  No  writer  upon 
such  subjects  can  expect  to  be  either  understood  or 
assented  to,  beyond  the  limits  of  a  narrow  school,  who 
is  not  prepared  with  good  arguments  as  well  as  magis 
terial  decisions  upon  the  controverted  points  of  philo 
sophy.  But  it  may  be  satisfactory  to  some  readers  to 
see  the  Philosophy,  to  which  in  the  present  work  we  are 
led,  presented  in  the  Aphoristic  form.  I  have  therefore 
placed  a  Series  of  Aphorisms  at  the  end  of  the  work. 
In  the  former  edition  these,  by  being  placed  at  the  begin 
ning  of  the  work,  might  mislead  the  reader ;  seeming 
to  some,  perhaps,  to  be  put  forwards  as  the  grounds,  not 
as  the  results,  of  our  philosophy.  I  have  also  prefixed 
an  analysis  of  the  work,  in  the  form  of  a  Table  of  Con 
tents  to  each  volume. 

In  that  part  of  the  second  volume  which  treats  of 
the  Language  of  Science,  I  have  made  a  few  alterations 
and  additions,  tending  to  bring  my  recommendations 
into  harmony  with  the  present  use  of  the  best  scientific 









Sect.  1.  Thoughts  and  Things. 

2.  Necessary  and  Experiential  Truths                           •  19 

3.  Deduction  and  Induction            .         .         .         .  .21 

4.  Theories  and  Facts 23 

5.  Ideas  and  Sensations          ......     24 

6.  Reflexion  and  Sensation          .....  27 
7-  Subjective  and  Objective     ...  .     29 

8.  Matter  and  Form 33 

9.  Man  the  Interpreter  of  Nature  .  .         .     37 

10.  The  Fundamental  Antithesis  is  inseparable     .         .         38 

11.  Successive  Generalization  .         .  .         .     46. 

CHAP.  III.     OF  TECHNICAL  TERMS  .         .        .         .        .        51 

Art.  1.     Examples. 

2.     Use  of  Terms. 


Art.  1.  The  two  Elements  of  Knowledge, 

2.  Shewn  by  necessary  Truths. 

3.  Examples  of  necessary  Truths  in  numbers. 

4.  The  opposite  cannot  be  distinctly  conceived. 

5.  Other  Examples. 

6.  Universal  Truths. 

CHAP.  V.     OF  EXPERIENCE  .  .  t>2 

Art.  1.     Experience  cannot  prove  necessary  Truths, 
2.     Except  when  aided  by  Ideas. 




Art,  1.     These  Grounds  are  Fundamental  Ideas. 

2.  These  are  to  be  reviewed. 

3.  Definitions  and  Axioms. 

4.  Syllogism, 

f).  Produces  no  new  Truths. 

(i.  Axioms  needed. 

7.  Axioms  depend  on  Ideas  : 

8.  So  do  Definitions. 

9.  Idea  not  completely  expressed. 


EXPERIENCE  .  .  74 

Art.  1.     No  connexion  observed. 

2.  Faculties  implied  in  observation. 

3.  We  arc  to  examine  our  Faculties. 

CHAP.  VIII.     OF  THE  PHILOSOPHY  OF  THE  SCIENCES      .  .     7^ 

Sciences  arranged  according  to  Ideas. 


CHAP.  I.     OF  THE  PURE  SCIENCES  .....         82 

Art.  I.     Geometry,  Arithmetic,  Algebra, 

2.  Are  not  Inductive  Sciences  : 

3.  Are  Mathematical  Sciences. 

4.  Mixed  Mathematics. 

5.  Space,  Time,  Number. 

CHAP.  II.     OF  THE  IDEA  OF  SPACE       .  84 

Art.  1.     Space  is  an  Idea, 

2.  Not  derived  from  Experience, 

3.  As  Geometrical  Truth  shews. 

4.  Space  is  a  Form  of  Experience. 

5.  The  phrase  not  essential. 

CHAP.  III.  OF  SOME  PECULIARITIES  OF  THE  IDEA  OF  SPACE'        .         88 

Art.  1.  Space  is  not  an  Abstract  Notion. 

2.  Space  is  infinite. 

3.  Space  is  real. 

4.  Space  is  a  Form  of  Intuition. 
f>.  Figure. 

0.     Three  Dimensions. 




SPACE .91 

Art.  I.     Geometry. 

2.  Definitions. 

3.  Axioms. 

4.  Not  Hypotheses. 

5.  Axioms  necessary. 

6.  Straight  lines. 
7-     Planes. 

8.     Elementary  Geometry. 


Art.  1.     How  is  Geometry  hypothetical? 

2.  What  was  Stewart's  view  ? 

3.  "  Legitimate  filiations  "  of  Definitions. 

4.  Is  a  Definition  a  complete  explanation  ? 

5.  Are  some  Axioms  Definitions  ? 
0.     Axiom  concerning  Circles. 

7.  Can  Axioms  become  truisms  ?  - 

8.  Use  of  such. 

CHAP.  VI.  OF  THE  PERCEPTION  OF  SPACE          .         .         .         .111 

Art.  1.  Which  Senses  apprehend  Space? 

2.  Perception  of  solid  figure. 

3.  Is  an  interpretation. 

4.  May  be  analysed. 
").  Outline. 

6.  Reversed  convexity. 

7-  Do  we  perceive  Space  by  Touch  ? 

8.  Brown's  Opinion. 

9.  The  Muscular  Sense. 

10.  Bell's  Opinion. 

11.  Perception  includes  Activity. 

12.  Perception  of  the  Skiey  Dome. 

13.  Reid's  Idomenians. 

14.  Motion  of  the  Eye. 

15.  Searching  Motion. 

16.  Sensible  Spot. 

17-  Expressions  implying  Motion. 

CHAP.  VII.  OF  THE  IDEA  OF  TIME     .  12"> 

Art.  1.  Time  an  Idea  not  derived  from  Experience. 

2.  Time  is  a  Form  of  Experience. 

VOL.  I.  W.  P. 



Art.  3.     Number. 

4.     Is  Time  derived  from  Motion  ? 


Art.  1.  Time  is  not  an  Abstract  Notion. 

2.  Time  is  infinite. 

3.  Time  is  a  Form  of  Intuition. 

4.  Time  is  of  one  Dimension, 

5.  And  no  more. 

6.  Rhythm. 

7-     Alternation. 
8.     Arithmetic. 

CHAP.  IX.  OF  THE  AXIOMS  WHICH  RELATE  TO  NUMBER     .         .     132 

Art.  I.  Grounds  of  Arithmetic. 

2.  Intuition. 

3.  Arithmetical  Axioms, 

4.  Are  Conditions  of  Numerical  Reasoning 

5.  In  all  Arithmetical  Operations. 

6.  Higher  Numbers. 

CHAP.  X.     OF  THE  PERCEPTION  OF  TIME  AND  NUMBER     .        .         135 
Art.  1.     Memory. 

2.  Sense  of  Successiveness 

3.  Implies  Activity. 

4.  Number  also  does  so. 

5.  And  apprehension  of  Rhythm. 

Note  to  Chapter  X      .         .         .         ...         .         .         .139 

CHAP.  XI.  OF  MATHEMATICAL  REASONING  .         .         .  141 

Art.  1.  Discursive  Reasoning. 

2.  Technical  Terms  of  Reasoning. 

3.  Geometrical  Analysis  and  Synthesis. 


Art.  1.  The  Idea  of  a  Limit. 

2.  The  use  of  General  Symbols. 

3.  Connexion  of  Symbols  and  Analysis. 

CHAP.  XIII.     THE  DOCTRINE  OF  MOTION      .         .         .         .  150 

Art.  1.     Pure  Mechanism. 
2.     Formal  Astronomy. 


INDUCTIVE  SCIENCES  .         .         .         .         .153 

Art.  1.     The  Ideas  of  Space  and  Number  are  clear  from  the 



Art.  2.  Their  application  in  Astronomy. 

3.  Conic  Sections,  &c. 

4.  Arabian  Numerals. 

5.  Newton's  Lemmas. 

6.  Tides. 

7.  Mechanics. 

8.  Optics. 

9.  Conclusion. 


CHAP.  I.     OF  THE  MECHANICAL  SCIENCES      .  .  164 

CHAP.  II.     OF  THE  IDEA  OF  CAUSE  .         .         .         .          .105 

Art.  1.  Not  derived  from  Observation. 

2.  As  appears  by  its  use. 

3.  Cause  cannot  be  observed. 

4.  Is  Cause  only  constant  succession  ? 

5.  Other  reasons. 


Art.  1.  Hume's  Doctrine. 

2.  Stewart  and  Brown. 

3.  Kant. 

4.  Relation  of  Kant  and  Brown. 

5.  Axioms  flow  from  the  Idea. 

6.  The  Idea  implies  activity  in  the  Mind. 


Art.  1.  Causes  are  Abstract  Conceptions. 

2.  First  Axiom. 

3.  Second  Axiom. 

4.  Limitation  of  the  Second  Axiom. 

5.  Third  Axiom. 

6.  Extent  of  the  Third  Axiom. 


MATTER         .  ....  185 

Art.  1.     Force. 

2.  Matter.  

3.  Solidity. 

4.  Inertia. 

5.  Application. 





STATICS      ....  .192 

Art.  1.  Object  of  the  Chapter. 

2.  Statics  and  Dynamics. 

3.  Equilibrium. 

4.  Measure  of  Statical  Forces. 

5.  The  Center  of  Gravity. 
(5.  Oblique  Forces. 

7-     Force  acts  at  any  point  of  its  Direction. 

8.  The  Parallelogram  of  Forces 


9.  Is  a  necessary  Truth. 

10.  Center  of  Gravity  descends. 

11.  Stevinus's  Proof. 

12.  Principle  of  Virtual  Velocities. 

13.  Fluids  press  equally. 

14.  Foundation  of  this  Axiom. 


DYNAMICS  ......  215 

Art.  1.  History. 

2.  The  First  Law  of  Motion. 

3.  Gravity  is  a  Uniform  Force. 

4.  The  Second  Law  of  Motion. 

5.  The  Third  Law  of  Motion. 

6.  Action  and  Reaction  in  Moving  Bodies. 
7-  D'Alembert's  Principle. 

8.  Connexion  of  Statics  and  Dynamics. 

9.  Mechanical  Principles  grow  more  evident. 
10.     Controversy  of  the  Measure  of  Force. 


OBTAINED  FROM  EXPERIENCE         ....       245 

Art.  1.  Experience  cannot  establish  necessary  Truths  ; 

2.  But  can  interpret  Axioms 

3.  Gives  us  the  Matter  of  Truths. 

4.  Exemplifies  Truths. 

5.  Cannot  shake  Axioms. 

6.  Is  this  applicable  in  other  cases  ? 


GRAVITATION          ......  254 

Art.  1.     General  course  of  the  History. 
2.      Particulars  as  to  the  Law. 



Art.  3.  As  to  the  Gravity  of  Matter. 

4.  Universality  of  the  Law. 

5.  Is  Gravity  an  essential  quality  ? 

6.  Newton's  Rule  of  Philosophizing. 
7-  Hypotheses  respecting  Gravity. 
8.  Do  Bodies  act  at  a  distance  ? 


IDEAS  ...  .  .  2(>2 

Art.  1 .  Nature  of  the  Process 

2.  Among  the  Ancients. 

3.  Kepler,  &c. 

4.  Lord  Monboddo,  &c. 

5.  Schelling,  &c. 
(i.  Common  usage. 

7.  Effect  of  Phrases. 

8.  Contempt  of  Predecessors. 

9.  Less  detail  hereafter. 

10.  Mechanico-Chemical  Sciences. 

11.  Secondary  Mechanical  Sciences. 

Additional  Note  to  Chapter  IV.     On  the  Axioms  which  relate  to 

the  Idea  of  Cause 274 

Additional  Note  to  Chapter  VI.  Sect.  5.   On  the  Center  of  Gravity   275 



Art.  1.     Of  Primary  and  Secondary  Qualities. 

2.  The  Idea  of  Externality. 

3.  Sensation  by  a  Medium. 

4.  Process  of  Perception  of  Secondary  Qualities. 

FERENT  SENSES  ...  .     28(i 

Art.  I.     Difference  of  Senses. 
Sect.  I.     Prerogatives  of  Sight. 
Art.  2.     Position. 
3.     Distance. 



Sect.  II.     Prerogatives  of  Hearing. 

Art.  4.  Musical  Intervals. 

5.  Chords. 

6.  Rhythm. 

Sect.  III.  The  Paradoxes  of  Vision. 

Art.  7-  First  Paradox. 

8.  Second  Paradox. 

9.  The  same  for  near  Objects. 
10.  Objections  answered. 

Sect.  IV.  The  Perception  of  Visible  Figures. 
Art.  11.     Brown's  Opinion. 

TION  OF  THE  IDEA  OF  A  MEDIUM         .         .         .     307 
Art.  1.     Introduction. 

2.  Sound. 

3.  Light. 

4.  Heat. 


Sect.  I.  Scales  of  Qualities  in  General. 
Art.  1.      Intensity. 

2.     Quantity  and  Quality. 

Sect.  II.  The  Musical  Scale. 

Art.  3.     Musical  Relations. 

4.     Musical  Standard. 

Sect.  III.     Scales  of  Colour. 
Art.  5.     The  Prismatic  Scale. 
6.     Newton's  Scale. 

7-  Scales  of  Impure  Colours. 
8.     Chromatometer. 

Sect.  IV.     Scales  of  Light. 
Art.  9.     Photometer. 
10.     Cyanometer. 

Sect.  V.  Scales  of  Heat. 
Art.  11.     Thermometers. 

12.  Their  progress. 

13.  Fixed  Points. 

14.  Concordance  of  Thermometers. 

15.  Natural  Measure. 



Art.  16.  Law  of  Cooling. 

17.  Theory  of  Exchanges. 

18.  Air  Thermometer. 

19.  Theory  of  Heat. 

20.  Other  Instruments. 

Sect.  VI.  Scales  of  other  Quantities. 
Art.  21.     Tastes  and  Smells. 

22.  Quality  of  Sounds. 

23.  Articulate  Sounds. 

24.  Transition. 




OF  POLARITY         .......     345 

Art.  1.     Introduction  of  the  Idea. 

2.  Magnetism. 

3.  Electricity. 

4.  Voltaic  Electricity. 

5.  Light. 

6.  Crystallization. 

7-     Chemical  Affinity. 

8.  General  Remarks. 

9.  Like  repels  like. 

CHAP.  II.     OF  THE  CONNEXION  OF  POLARITIES          .        .         .    357 

Art.  1.  Different  Polar  Phenomena  from  one  Cause. 

2.  Connexion  of  Magnetic  and  Electric  Polarity. 

3.  Ampere's  Theory. 

4.  Faraday's  views. 

5.  Connexion  of  Electrical  and  Chemical  Polarity. 

6.  Davy's  and  Faraday's  views 

7«     Depend  upon  Ideas  as  well  as  Experiments. 

8.  Faraday's  Anticipations. 

9.  Connexion  of  Chemical  and  Crystalline  Polarities. 

10.  Connexion  of  Crystalline  and  Optical  Polarities. 

11.  Connexion  of  Polarities  in  general. 

12.  Schelling's  Speculations. 

13.  Hegel's  vague  notions. 

14.  Ideas  must  giiide  Experiment. 





Art.  1.  Fundamental  Ideas  of  Chemistry. 

2.  Elements. 

3.  Do  Compounds  resemble  their  Elements  ? 

4.  The  Three  Principles. 

5.  A  Modern  Errour. 

6.  Are  Compounds  determined  by  the  Figure  of  Ele 

ments  ? 

7.  Crystalline  Form  depends  on  Figure  of  Elements. 

8.  Are  Compounds  determined  by  Mechanical  Attrac 

tion  of  Elements  ? 

9.  Newton's  followers. 

10.     Imperfection  of  their  Hypotheses. 

CHEMICAL  AFFINITY    ....          .         .     38! 

Art.  1.     Early  Chemists. 

2.  Chemical  Affinity. 

3.  Affinity  or  Attraction  ? 

4.  Affinity  preferable. 

5.  Analysis  is  possible. 

6.  Affinity  is  Elective. 
7-     Controversy  on  this. 

8.  Affinity  is  Definite. 

9.  Are  these  Principles  necessarily  true  ? 

10.  Composition  determines  Properties. 

11.  Comparison  on  this  subject. 

12.  Composition  determines  Crystalline  Form. 

CHAP.  III.  OF  THE  IDEA  OF  SUBSTANCE  ....     404 

Art.  1.     Indestructibility  of  Substance. 

2.  The  Idea  of  Substance. 

3.  Locke's  Denial  of  Substance. 

4.  Is  all  Substance  heavy  ? 


MISTKY       . 412 

Art.  1.     A  Body  is  Equal  to  its  Elements. 

2.  Lavoisier. 

3.  Are  there  Imponderable  Elements  ? 



Art.  4.     Faraday's  views. 

5.  Composition  of  Water. 

6.  Heat  in  Chemistry. 

CHAP.  V.     THE  ATOMIC  THEORY      .  .421 

Art.  1.  The  Theory  on  Chemical  Grounds. 

2.  Hypothesis  of  Atoms. 

3.  Its  Chemical  Difficulties. 

4.  Grounds  of  the  Atomic  Doctrine. 

5.  Ancient  Atomists. 
(5.  Francis  Bacon. 

7-  Modern  Atomists. 

8.  Arguments  for  and  against. 

I).  Boscovich's  Theory. 

10.  Molecular  Hypothesis. 

11.  Poisson's  Inference. 

12.  Wollaston's  Argument. 

13.  Properties  are  Permanent. 




Art.  1.  Symmetry  what. 

2.  Kinds  of  Symmetry. 

3.  Examples  in  Nature. 

4.  Vegetables  and  Animals. 

5.  Symmetry  a  Fundamental  Idea. 
0.  Result  of  Symmetry. 

Art.  1.     "  Fundamental  Forms." 

2.  Their  use. 

3.  "  Systems  of  Crystallization." 

4.  Cleavage. 

5.  Other  Properties. 


CRYSTALS  ...  .     452 

Art.  1.     Integrant  Molecules 

2.  Difficulties  of  the  Theory. 

3.  Merit  of  the  Theory. 

4.  Wollaston's  Hypothesis. 



Art.  5.  Maxim  for  such   Hypotheses. 

6.  Dalton's  Hypothesis. 

7-  Ampere's  Hypothesis. 

8.  Difficulty  of  such   Hypotheses. 

9.  Isomorphism. 



COMMON  NAMES  ......     466 

Art.  1.  Object  of  the  Chapter. 

2.  Unity  of  the  Individual. 

3.  Condition  of  Unity. 

4.  Kinds. 

5.  Not  made  by  Definitions. 

6.  Condition  of  the  Use  of  Terms. 
7-  Terms  may  have  different  Uses. 

8.  Gradation  of  Kinds. 

9.  Characters  of  Kinds. 

10.  Difficulty  of  Definitions. 

11.  "  The  Five  Words." 



,.    Sect.  I.     Natural   History  in   General. 

Art.  1.     Idea  of  Likeness  in  Natural  History. 
2.     Condition  of  its  Use. 

Sect.  II.      Terminology. 

Art.  3.     Meaning  of  the  word. 

The  Plan  of  the  System, 

Its  Meaning. 

Latent  Reference  to  Natural  Affinity. 

Natural  Classes. 

Artificial  Classes. 

Are  Genera  Natural? 

Natural  History  and  Mathematics. 

Natural  Groups  given  by  Type,  not  by  Definition. 


Artificial  and  Natural  Systems. 



Sect.  IV.     Methods  of  framing  Natural  Systems. 

Art.  13.     Method  of  Blind  Trial. 

14.     Method  of  General  Comparison. 

Sect.  V.     Gradation  of  Groups. 

Art.  15.     Series  of  Subdivisions. 
What  is  a  Species  ? 
The  words  "  Species"  and  "  Genus." 

Varieties.     Races. 

Sect.  VI.  Nomenclature. 

Art.  19.  Binary  Nomenclature. 
Sect.  VII.     Diagnosis. 

Art.  20.  Characteristick  and  Systematick. 


MINERALOGY       .  *  .        .     512 

Art.  1.  Mohs's  System. 

2.  His  "  Characteristick." 

3.  Mineral  Species  not  yet  well  fixed. 

4.  Orders  of  Minerals. 

5.  Nomenclature  of  Minerals. 

6.  M.  Necker's  "  Rcgne  Mineral." 

7-     Inconvenience  of  taking  a  Chemical  basis  of  Mineral 

8.  Relation  of  Natural  History  and  Chemistry. 

9.  What  is  a  Mineralogical  Individual  ? 

10.  A  well-formed  Crystal  is  an  Individual. 

11.  Not  the  Integrant  Molecules, 

12.  Nor  the  Cleavage  Forms. 

13.  Compound  Crystals  are  not  individuals. 

14.  Crystalline  Forms  are  sufficiently  complete  for  this. 

15.  Including  aggregate  Masses. 

16.  Do  Artificial  Crystals  belong  to  Mineralogy  ? 

17.  The    Mineralogical   Individual   extends   as  far  as 

the  same  Crystalline  Axes  extend. 

18.  Artificial  Crystals  do  belong  to  Mineralogy : 

19.  Cannot  be  excluded. 

20.  Species  to  be  determined  by  the  Crystalline  Power. 

21.  Secondary  Derivative  Forms  are  Varieties: 

22.  Are  not  Species,  as  M.  Necker  holds. 




Art.  1.     The  Idea  of  Affinity 

2.  Is  not  to  be  made  out  by  Arbitrary  Eules. 

3.  Functions  of  Living  tilings  are  many, 

4.  But  all  lead  to  the  same  arrangement. 

5.  This  is  Cuvier's  principle  : 
(5.  And  Decandolle's. 

7-      Is  this  applicable  to  Inorganic  Bodies  ? 
8.     Yes ;  by  the  agreement  of  Physical   and  Chemical 

BOOK    IX. 


Art.  1 .     Biology  involves  the  Idea  of  Life. 


2.  This  Idea  to  be  historically  traced. 

3.  The  Idea  at  first  expressed  by  means  of  other  Ideas, 

4.  Mystical,  Mechanical,   Chemical,  and  Vital   Fluid 


CHAP.  II.     SUCCESSIVE  BIOLOGICAL  HYPOTHESES     .         .         .  548 

Sect.  I.      The  Mystical  School. 
Sect.  II.      The  latrochemical  School. 
Sect.  III.     The  latromathematical  School. 
Sect.  IV.     The  Vitnl  Fluid  School. 
Sect.  V.      The  Psychical  School. 

CHAP.  III.     ATTEMPTS  TO  ANALYSE  THE  IDEA  OF  LIFE         .         .     571 

A  rt.  } .  Definitions  of  Life, 

2.  By  Stahl,  Humboldt,  Kant. 

3.  Definition  of  Organization  by  Kant. 

4.  Life  is  a  System  of  Functions. 

5.  Bichat.     Sum  of  Functions. 
<).  Use  of  Definition. 

7-  Cuvier's  view. 

8.  Classifications  of  Functions. 

0.  Vital,  Natural,  and  Animal  Functions. 

10.  Bichat.      Organic  and  Animal  Life. 

1  1 .  Use  of  this  Classification. 




TION  580 

Sect.  I.     Course  of  Biological  Research. 

Art.  1.     Observation  and  New  Conceptions. 

Sect.  II.  Attempts  to  form  a  distinct  Conception  of  Assimila 
tion  and  Secretion. 
Art.  2.     The  Ancients. 

3.  Buffon.     Interior  Mould. 

4.  Defect  of  this  view. 

5.  Cuvier.     Life  a  Vortex. 

6.  Defect  of  this  view. 

7.  Schclling.     Matter  and  Form. 

8.  Life  a  constant  Form  of  circulating  Matter,  &c. 

Sect.  III.  Attempts  to  conceive  the  Forces  of  Assimilation  and 


Art.  9.  Assimilation  is  a  Vital  Force. 

10.  The  name  "Assimilation." 

11.  Several  processes  involved  in  Assimilation. 

12.  Absorption.     Endosmose. 

13.  Absorption  involves  a  Vital  Force. 

14.  Secretion.     Glands. 

15.  Motions  of  Vital  Fluids. 

Sect.  IV.  Attempts  to  conceive  the  Process  of  Generation. 

Art.  16.  Reproduction  figuratively  used  for  Generation. 

1 7-  Nutrition  different  from 

18.  Generation. 

19.  Generations  successively  included. 

20.  Pre-existence  of  Germs. 

21.  Difficulty  of  this  view. 

22.  Communication  of  Vital  Forces. 

23.  Close  similarity  of  Nutrition  and  Generation. 

24.  The  Identity  of  the  two  Processes  exemplified. 


continued. — VOLUNTARY  MOTION       .         .         .  (iOO 

Art.  1.  Voluntary  Motion  one  of  the  animal  Functions. 

2.  Progressive  knowledge  of  it. 

3.  Nervous  Fluid  not  electric. 

4.  Irritability.     Glisson. 

5.  Ilaller. 



Art.  6.  Contractility. 

7.  Organic  Sensibility  and  Contractility  not  separable. 

8.  Improperly  described  by  Bichat. 

9.  Brown. 

10.  Contractility  a  peculiar  Power. 

11.  Cuvier's  view. 

12.  Elementary  contractile  Action. 

13.  Strength  of  Muscular  Fibre. 

14.  Sensations  become  Perceptions 

15.  By  means  of  Ideas  ; 

16.  And  lead  to  Muscular  Actions. 

17-  Volition  comes  between  Perception  and  Action. 

18.  Transition  to  Psychology. 

19.  A  center  is  introduced. 

20.  The  central  consciousness  may  be  obscure. 

21.  Reflex  Muscular  Action. 

22.  Instinct. 

23.  Difficulty  of  conceiving  Instinct. 

24.  Instinct  opposed  to  Insight. 

CHAP.  VI.  OF  THE  IDEA  OF  FINAL  CAUSES    .         .         .  618 

Art.  1.  Organization.     Parts  are  Ends  and  Means. 

2.  Not  merely  mutually  dependent. 

3.  Not  merely  mutually  Cause  and  Effect. 

4.  Notion  of  End  not  derived  from  Facts. 

5.  This  notion  has  regulated  Physiology. 

6.  Notion  of  Design  comes  from  within. 
7-  Design  not  understood  by  Savages. 

8.  Design  opposed  to  Morphology. 

9.  Impression  of  Design  when  fresh. 

10.  Acknowledgement  of  an  End  by  adverse  Physiolo 


1 1 .  This  included  in  the  Notion  of  Disease. 

12.  It  belongs  to  Organized  Creatures  only. 

13.  The  term  Final  Cause- 

14.  Law  and  Design. 

15.  Final  Causes  and  Morphology. 

16.  Expressions  of  physiological  Ends. 
17-  The  Conditions  of  Existence. 

18.  The  asserted  presumption  of  Teleology. 

19.  Final  Causes  in  other  subjects. 

20.  Transition  to  Palaetiology. 



BOOK   X. 


CHAP.  I.     OF  PALETIOLOGICAL  SCIENCES  IN  GENERAL        .         .     C37 
Art.  1.     Description  of  Paltetiology. 

2.  Its  Members. 

3.  Other  Members. 

4.  Connexion  of  the  whole  subject. 

5.  We  shall  take  Material  Sciences  only; 

6.  But  these  are  connected  with  others. 


SCIENCE      ........     642 

Art.  1.     Divisions  of  such  Sciences. 

2.  The  Study  of  Causes. 

3.  vEtiology. 

4.  Phenomenology  requires  Classification.    Phenomenal 


5.  Phenomenal  Uranology. 

6.  Phenomenal  Geography  of  Plants  and  Animals. 
7-      Phenomenal  Glossology. 

8.  The  Study  of  Phenomena  leads  to  Theory. 

9.  No  sound  Theory  without  ^Etiology. 

10.  Causes  in  Palietiology. 

11.  Various  kinds  of  Cause. 

12.  Hypothetical  Order  of  Palsetiological  Causes. 

13.  Mode  of  Cultivating  ./Etiology  : — In  Geology  : 

14.  In  the  Geography  of  Plants  and  Animals : 

15.  In  Languages. 

16.  Construction  of  Theories. 

17.  No  sound  Paleetiological  Theory  yet  extant. 

TRINE  OF  UNIFORMITY        .  ...     665 
Art.  1.     Doctrine  of  Catastrophes. 

2.  Doctrine  of  Uniformity. 

3.  Is  Uniformity  probable  a  priori  ? 

4.  Cycle  of  Uniformity  indefinite. 

5.  Uniformitarian  Arguments  are  Negative  only. 

6.  Uniformity   in  the  Organic  World. 

7-     Origin  of  the  present  Organic  World. 

8.  Nebular  Origin  of  the  Solar  System. 

9.  Origin  of  Languages. 

10.     No  Natural  Origin  discoverable. 




Art.  1.     Importance  of  Tradition. 

2.  Connexion  of  Tradition  and  Science. 

3.  Natural  and  Providential  History   of  the  World. 

4.  The  Sacred  Narrative. 

5.  Difficulties  in  interpreting  the  Sacred  Narrative, 
(x      Such  Difficulties  inevitable. 

7.  Science  tells  us  nothing  concerning  Creation. 

8.  Scientific  views,  when  familiar,  do  not  disturb  the 

authority  of  Scripture. 

9.  When  should  Old  Interpretations  be  given  up  ? 

10.  In  what  Spirit  should  the  Change  be  accepted  ? 

11.  In  what  Spirit  should  the  Change  be  urged? 

12.  Duty  of  Mutual  Forbearance. 

13.  Case  of  Galileo. 

CHAP.  V.     OF  THE  CONCEPTION  OF  A  FIRST  CAUSE    .         .         .     700 

Art.  I.     The  Origin  of  things  is  not  naturally  discoverable; 

2.  Yet  has  always  been  sought  after. 

3.  There  must  be  a  First  Cause. 

4.  This  is  an  Axiom. 

5.  Involved  in  the  Proof  of  a  Deity. 

(?.  The  Mind  is  not  satisfied  without  it. 

7-  The  Whole  Course  of  Nature  must  have  a  Cause. 

8.  Necessary  Existence  of  God. 

9.  Forms  of  the  Proof. 

10.  Idea  of  a  First  Cause  is  Necessary. 

11.  Conception  of  a  First  Cause. 

12.  The  First  Cause  in  all  Sciences  is  the  same. 

13.  We  are  thus  led  to  Moral  Subjects. 
Conclusion  of  Part  I. 





PART   I. 


VOL.  I.      W.  P. 

l^uu,'  adhuc  inventa  sunt  in  Scieutiis,  ea  hujusiaodi  sunt 
nt  Notionibus  Vnlgaribus  fere  subjaceant :  ut  vero  ad  inte- 
riora  ct  remotiora  Nature  pcnetrctur,  uecesse  cst  ut  tarn 
NoxiOiNEs  quani  AXIOMATA  inagis  certa  et  nninita  \ia  a 
particularibus  abstrahantur ;  atque  omnino  molior  ct  certior 
iiitellcctus  adoperatio  in  usum  veniat. 

BACON,  Nov.  Org.,  Lib.  i.  Aphor.  xviii. 




THE  PHILOSOPHY  OF  SCIENCE,  if  the  phrase  were  to  be 
understood  in  the  comprehensive  sense  which  most  na 
turally  offers  itself  to  our  thoughts,  would  imply  nothing 
less  than  a  complete  insight  into  the  essence  and  con 
ditions  of  all  real  knowledge,  and  an  exposition  of  the 
best  methods  for  the  discovery  of  new  truths.  We  must 
narrow  and  lower  this  conception,  in  order  to  mould  it 
into  a  form  in  which  we  may  make  it  the  immediate 
object  of  our  labours  with  a  good  hope  of  success ;  yet 
still  it  may  be  a  rational  and  useful  undertaking,  to 
endeavour  to  make  some  advance  towards  such  a  Philo 
sophy,  even  according  to  the  most  ample  conception 
of  it  which  we  can  form.  The  present  work  has  been 
written  with  a  view  of  contributing,  in  some  measure, 
however  small  it  may  be,  towards  such  an  undertaking. 

But  in  this,  as  in  every  attempt  to  advance  beyond 
the  position  which  we  at  present  occupy,  our  hope  of 
success  must  depend  mainly  upon  our  being  able  to 
profit,  to  the  fullest  extent,  by  the  progress  already 
made.  We  may  best  hope  to  understand  the  nature  and 
conditions  of  real  knowledge,  by  studying  the  nature 
and  conditions  of  the  most  certain  and  stable  portions  of 
knowledge  which  we  already  possess :  and  we  are  most 
likely  to  learn  the  best  methods  of  discovering  truth,  by 
VOL.  i.  w.  P.  B 

2  OF    IDEAS    IN    GENERAL. 

examining  how  truths,  now  universally  recognized,  have 
really  been  discovered.  Now  there  do  exist  among  us 
doctrines  of  solid  and  acknowledged  certainty,  and 
truths  of  which  the  discovery  has  been  received  with 
universal  applause.  These  constitute  what  we  com 
monly  term  Sciences ;  and  of  these  bodies  of  exact  and 
enduring  knowledge,  we  have  within  our  reach  so  large 
and  varied  a  collection,  that  we  may  examine  them,  and 
the  history  of  their  formation,  with  a  good  prospect  of 
deriving  from  the  study  such  instruction  as  we  seek. 
We  may  best  hope  to  make  some  progress  towards  the 
Philosophy  of  Science,  by  employing  ourselves  upon  THE 

The  Sciences  to  which  the  name  is  most  commonly 
and  unhesitatingly  given,  are  those  which  are  concerned 
about  the  material  world ;  whether  they  deal  with  the 
celestial  bodies,  as  the  sun  and  stars,  or  the  earth  and 
its  products,  or  the  elements ;  whether  they  consider  the 
differences  which  prevail  among  such  objects,  or  their 
origin,  or  their  mutual  operation.  And  in  all  these 
Sciences  it  is  familiarly  understood  and  assumed,  that 
their  doctrines  are  obtained  by  a  common  process  of 
collecting  general  truths  from  particular  observed  facts, 
which  process  is  termed  Induction.  It  is  further  assumed 
that  both  in  these  and  in  other  provinces  of  knowledge, 
so  long  as  this  process  is  duly  and  legitimately  per 
formed,  the  results  will  be  real  substantial  truth.  And 
although  this  process,  with  the  conditions  under  which 
it  is  legitimate,  and  the  general  laws  of  the  formation  of 
Sciences,  will  hereafter  be  subjects  of  discussion  in  this 
work,  I  shall  at  present  so  far  adopt  the  assumption  of 
which  I  speak,  as  to  give  to  the  Sciences  from  which 
our  lessons  are  to  be  collected  the  name  of  Inductive 
Sciences.  And  thus  it  is  that  I  am  led  to  designate  my 


The   views  respecting  the  nature  and  progress   of 
knowledge,  towards  which  we  shall  be  directed  by  such 
a  course  of  inquiry  as  I  have  pointed  out,  though  derived 
from  those  portions   of  human  knowledge  which  are 
more  peculiarly  and  technically  termed  Sciences,  will  by 
no  means  be  confined,  in  their  bearing,  to  the  domain  of 
such  Sciences  as  deal  with  the  material  world,  nor  even 
to  the  whole  range  of  Sciences  now  existing.     On  the 
contrary,  we  shall  be  led  to  believe  that  the  nature  of 
truth  is  in  all  subjects  the  same,  and  that  its  discovery 
involves,  in  all  cases,  the  like  conditions.     On  one  sub 
ject  of  human  speculation  after  another,  man's  know 
ledge  assumes  that  exact  and  substantial  character  which 
leads  us  to  term  it  Science ;  and  in  all  these  cases,  whe 
ther  inert  matter  or  living  bodies,  whether  permanent 
relations  or  successive  occurrences,  be  the  subject  of  our 
attention,  we  can  point  out  certain  universal  characters 
which  belong  to  truth,  certain  general  laws  which  have 
regulated  its  progress  among  men.     And  we  naturally 
expect  that,  even  when  we  extend  our  range  of  specu 
lation  wider  still,  when  we  contemplate  the  world  within 
us  as  well  as  the  world  without  us,  when  we  consider 
the  thoughts  and  actions  of  men  as  well  as  the  motions 
and  operations  of  unintelligent  bodies,  we  shall  still  find 
some  general  analogies  which  belong  to  the  essence  of 
truth,  and  run  through  the  whole  intellectual  universe. 
Hence  we  have  reason  to  trust  that  a  just  Philosophy  of 
the  Sciences  may  throw  light  upon  the  nature  and  extent 
of  our  knowledge  in  every  department  of  human  specu 
lation.     By  considering  what  is  the  real  import  of  our 
acquisitions,  where  they  are  certain  and  definite,  we  may 
learn  something  respecting  the  difference  between  true 
knowledge  and  its  precarious  or  illusory  semblances ;  by 
examining  the  steps  by  which  such  acquisitions  have 
been  made,  we  may  discover  the  conditions  under  which 

B  12 

4  OF    IDEAS    IN    GENERAL. 

truth  is  to  be  obtained ;  by  tracing  the  boundary-line 
between  our  knowledge  and  our  ignorance,  we  may 
ascertain  in  some  measure  the  extent  of  the  powers  of 
man's  understanding. 

But  it  may  be  said,  in  such  a  design  there  is  nothing 
new;  these  are  objects  at  which  inquiring  men  have 
often  before  aimed.  To  determine  the  difference  be 
tween  real  and  imaginary  knowledge,  the  conditions 
under  which  we  arrive  at  truth,  the  range  of  the  powers 
of  the  human  mind,  has  been  a  favourite  employment  of 
speculative  men  from  the  earliest  to  the  most  recent 
times.  To  inquire  into  the  original,  certainty,  and  com 
pass  of  man's  knowledge,  the  limits  of  his  capacity,  the 
strength  and  weakness  of  his  reason,  has  been  the  pro 
fessed  purpose  of  many  of  the  most  conspicuous  and 
valued  labours  of  the  philosophers  of  all  periods  up  to 
our  own  day.  It  may  appear,  therefore,  that  there  is 
little  necessity  to  add  one  more  to  these  numerous 
essays ;  and  little  hope  that  any  new  attempt  will  make 
any  very  important  addition  to  the  stores  of  thought 
upon  such  questions,  which  have  been  accumulated  by 
the  profoundest  and  acutest  thinkers  of  all  ages. 

To  this  I  reply,  that  without  at  all  disparaging  the 
value  or  importance  of  the  labours  of  those  who  have 
previously  written  respecting  the  foundations  and  con 
ditions  of  human  knowledge,  it  may  still  be  possible  to 
add  something  to  what  they  have  done.  The  writings  of 
all  great  philosophers,  up  to  our  own  time,  form  a  series 
which  is  not  yet  terminated.  The  books  and  systems  of 
philosophy  which  have,  each  in  its  own  time,  won  the 
admiration  of  men,  and  exercised  a  powerful  influence 
upon  their  thoughts,  have  had  each  its  own  part  and 
functions  in  the  intellectual  history  of  the  world ;  and 
other  labours  which  shall  succeed  these  may  also  have 
their  proper  office  and  useful  effect.  We  may  not  be 


able  to  do  much,  and  yet  still  it  may  be  in  our  power  to 
effect  something.  Perhaps  the  very  advances  made  by 
former  inquirers  may  have  made  it  possible  for  us,  at 
present,  to  advance  still  further.  In  the  discovery  of 
truth,  in  the  developement  of  man's  mental  powers  and 
privileges,  each  generation  has  its  assigned  part ;  and  it 
is  for  us  to  endeavour  to  perform  our  portion  of  this 
perpetual  task  of  our  species.  Although  the  terms 
which  describe  our  undertaking  may  be  the  same  which 
have  often  been  employed  by  previous  writers  to  express 
their  purpose,  yet  our  position  is  different  from  theirs, 
and  thus  the  result  mav  be  different  too.  We  have,  as 


they  had,  to  run  our  appropriate  course  of  speculation 
with  the  exertion  of  our  best  powers ;  but  our  course 
lies  in  a  more  advanced  part  of  the  great  line  along 
which  Philosophy  travels  from  age  to  age.  However 
familiar  and  old,  therefore,  be  the  design  of  such  a  work 
as  this,  the  execution  may  have,  and  if  it  be  performed 
in  a  manner  suitable  to  the  time,  will  have,  something 
that  is  new  and  not  unimportant. 

Indeed,  it  appears  to  be  absolutely  necessary,  in 
order  to  check  the  prevalence  of  grave  and  pernicious 
errour,  that  the  doctrines  which  are  taught  concerning 
the  foundations  of  human  knowledge  and  the  powers  of 
the  human  mind,  should  be  from  time  to  time  revised 
and  corrected  or  extended.  Erroneous  and  partial  views 
are  promulgated  and  accepted ;  one  portion  of  the  truth 
is  insisted  upon  to  the  undue  exclusion  of  another ;  or 
principles  true  in  themselves  are  exaggerated  till  they 
produce  on  men's  minds  the  effect  of  falsehood.  When 
evils  of  this  kind  have  grown  to  a  serious  height,  a 
Reform  is  requisite.  The  faults  of  the  existing  systems 
must  be  remedied  by  correcting  what  is  wrong,  and  sup 
plying  what  is  wanting.  In  such  cases,  all  the  merits 
and  excellencies  of  the  labours  of  the  preceding  times  do 

6  OF    IDEAS    IN    GENERAL. 

not  supersede  the  necessity  of  putting  forth  new  views 
suited  to  the  emergency  which  has  arrived.  The  new 
form  which  errour  has  assumed  makes  it  proper  to 
endeavour  to  give  a  new  and  corresponding  form  to 
truth.  Thus  the  mere  progress  of  time,  and  the  natural 
growth  of  opinion  from  one  stage  to  another,  leads  to 
the  production  of  new  systems  and  forms  of  philosophy. 
It  will  be  found,  I  think,  that  some  of  the  doctrines  now 
most  widely  prevalent  respecting  the  foundations  and 
nature  of  truth  are  of  such  a  kind  that  a  Reform  is 
needed.  The  present  age  seems,  by  many  indications,  to 
be  called  upon  to  seek  a  sounder  Philosophy  of  Know 
ledge  than  is  now  current  among  us.  To  contribute 
towards  such  a  Philosophy  is  the  object  of  the  present 
work.  The  work  is,  therefore,  like  all  works  which 
take  into  account  the  most  recent  forms  of  speculative 
doctrine,  invested  with  a  certain  degree  of  novelty  in  its 
aspect  and  import,  by  the  mere  time  and  circumstances 
of  its  appearance. 

But,  moreover,  we  can  point  out  a  very  important 
peculiarity  by  which  this  work  is,  in  its  design,  distin 
guished  from  preceding  essays  on  like  subjects ;  and  this 
difference  appears  to  be  of  such  a  kind  as  may  well 
entitle  us  to  expect  some  substantial  addition  to  our 
knowledge  as  the  result  of  our  labours.  The  peculiarity 
of  which  I  speak  has  already  been  announced ; — it  is 
this  :  that  we  purpose  to  collect  our  doctrines  concerning 
the  nature  of  knowledge,  and  the  best  mode  of  acquiring 
it,  from  a  contemplation  of  the  Structure  and  History  of 
those  Sciences  (the  Material  Sciences),  which  are  univer 
sally  recognized  as  the  clearest  and  surest  examples  of 
knowledge  and  of  discovery.  It  is  by  surveying  and 
studying  the  whole  mass  of  such  Sciences,  and  the 
various  steps  of  their  progress,  that  we  now  hope  to 
approach  to  the  true  Philosophy  of  Science. 


Now  this,  I  venture  to  say,  is  a  new  method  of  pur 
suing  the  philosophy  of  human  knowledge.  Those  who 
have  hitherto  endeavoured  to  explain  the  nature  of 
knowledge,  and  the  process  of  discovery,  have,  it  is  true, 
often  illustrated  their  views  by  adducing  special  exam 
ples  of  truths  which  they  conceived  to  be  established, 
and  by  referring  to  the  mode  of  their  establishment. 
But  these  examples  have,  for  the  most  part,  been  taken 
at  random,  not  selected  according  to  any  principle  or 
system.  Often  they  have  involved  doctrines  so  pre 
carious  or  so  vague  that  they  confused  rather  than  eluci 
dated  the  subject ;  and  instead  of  a  single  difficulty,— 
What  is  the  nature  of  Knowledge?  these  attempts  at 
illustration  introduced  two, — What  was  the  true  analysis 
of  the  Doctrines  thus  adduced?  and, — Whether  they 
might  safely  be  taken  as  types  of  real  Knowledge  ? 

This  has  usually  been  the  case  when"  there  have 
been  adduced,  as  standard  examples  of  the  formation  of 
human  knowledge,  doctrines  belonging  to  supposed  sci 
ences  other  than  the  material  sciences;  doctrines,  for 
example,  of  Political  Economy,  or  Philology,  or  Morals, 
or  the  Philosophy  of  the  Fine  Arts.  I  am  very  far  from 
thinking  that,  in  regard  to  such  subjects,  there  are  no 
important  truths  hitherto  established :  but  it  would  seem 
that  those  truths  which  have  been  obtained  in  these 
provinces  of  knowledge,  have  not  yet  been  fixed  by 
means  of  distinct  and  permanent  phraseology,  and  sanc 
tioned  by  universal  reception,  and  formed  into  a  con 
nected  system,  and  traced  through  the  steps  of  their 
gradual  discovery  and  establishment,  so  as  to  make  them 
instructive  examples  of  the  nature  and  progress  of  truth 
in  general.  Hereafter  we  trust  to  be  able  to  show  that 
the  progress  of  moral,  and  political,  and  philological, 
and  other  knowledge,  is  governed  by  the  same  laws  as 
that  of  physical  science.  But  since,  at  present,  the 

8  OF    IDEAS    IN    GENERAL. 

former  class  of  subjects  are  full  of  controversy,  doubt, 
and  obscurity,  while  the  latter  consist  of  undisputed 
truths  clearly  understood  and  expressed,  it  may  be  con 
sidered  a  wise  procedure  to  make  the  latter  class  of 
doctrines  the  basis  of  our  speculations.  And  on  the 
having  taken  this  course,  is,  in  a  great  measure,  my 
hope  founded,  of  obtaining  valuable  truths  which  have 
escaped  preceding  inquirers. 

But  it  may  be  said  that  many  preceding  writers  on 
the  nature  and  progress  of  knowledge  have  taken  their 
examples  abundantly  from  the  Physical  Sciences.  It 
would  be  easy  to  point  out  admirable  works,  which  have 
appeared  during  the  present  and  former  generations,  in 
which  instances  of  discovery,  borrowed  from  the  Phy 
sical  Sciences,  are  introduced  in  a  manner  most  happily 
instructive.  And  to  the  works  in  which  this  has  been 
done,  I  gladly  give  my  most  cordial  admiration.  But  at 
the  same  time  I  may  venture  to  remark  that  there  still 
remains  a  difference  between  my  design  and  theirs :  and 
that  I  use  the  Physical  Sciences  as  exemplifications  of 
the  general  progress  of  knowledge  in  a  manner  very 
materially  different  from  the  course  which  is  followed  in 
works  such  as  are  now  referred  to.  For  the  conclusions 
stated  in  the  present  work,  respecting  knowledge  and 
discovery,  are  drawn  from  a  connected  and  systematic 
survey  of  the  whole  range  of  Physical  Science  and  its 
History ;  whereas,  hitherto,  philosophers  have  contented 
themselves  with  adducing  detached  examples  of  scientific 
doctrines,  drawn  from  one  or  two  departments  of  science. 
So  long  as  we  select  our  examples  in  this  arbitrary  and 
limited  manner,  we  lose  the  best  part  of  that  philosophi 
cal  instruction,  which  the  sciences  are  fitted  to  afford 
when  we  consider  them  as  all  members  of  one  series, 
and  as  governed  by  rules  which  are  the  same  for  all. 
Mathematical  and  chemical  truths,  physical  and  physio- 


logical  doctrines,  the  sciences  of  classification  and  of 
causation,  must  alike  be  taken  into  our  account,  in  order 
that  we  may  learn  what  are  the  general  characters  of 
real  knowledge.  When  our  conclusions  assume  so  com 
prehensive  a  shape  that  they  apply  to  a  range  of  sub 
jects  so  vast  and  varied  as  these,  we  may  feel  some  con 
fidence  that  they  represent  the  genuine  form  of  universal 
and  permanent  truth.  But  if  our  exemplification  is  of  a 
narrower  kind,  it  may  easily  cramp  and  disturb  our  phi 
losophy.  We  may,  for  instance,  render  our  views  of 
truth  and  its  evidence  so  rigid  and  confined  as  to  be 
quite  worthless,  by  founding  them  too  much  on  the  con 
templation  of  mathematical  truth.  We  may  overlook 
some  of  the  most  important  steps  in  the  general  course 
of  discovery,  by  fixing  our  attention  too  exclusively 
upon  some  one  conspicuous  group  of  discoveries,  as,  for 
instance,  those  of  Newton.  We  may  misunderstand  the 
nature  of  physiological  discoveries,  by  attempting  to 
force  an  analogy  between  them  and  discoveries  of  me 
chanical  laws,  and  by  not  attending  to  the  intermediate 
sciences  which  fill  up  the  vast  interval  between  these 
extreme  terms  in  the  series  of  material  sciences.  In 
these  and  in  many  other  ways,  a  partial  and  arbitrary 
reference  to  the  material  sciences  in  our  inquiry  into 
human  knowledge  may  mislead  us ;  or  at  least  may  fail 
to  give  us  those  wider  views,  and  that  deeper  insight, 
which  should  result  from  a  systematic  study  of  the  whole 
range  of  sciences  with  this  particular  object. 

The  design  of  the  following  work,  then,  is  to  form  a 
Philosophy  of  Science,  by  analyzing  the  substance  and 
examining  the  progress  of  the  existing  body  of  the  sci 
ences.  As  a  preliminary  to  this  undertaking,  a  survey 
of  the  history  of  the  sciences  was  necessary.  This, 
accordingly,  I  have  already  performed ;  and  the  result 
of  the  labour  thus  undertaken  has  been  laid  before  the 
public  as  a  History  of  the  I?nluctire  Sciences. 

10  OF    IDEAS   IX    GEXERAL. 

In  that  work  I  have  endeavoured  to  trace  the  steps 
by  which  men  acquired  each  main  portion  of  that  know 
ledge  on  which  they  now  look  with  so  much  confidence 
and  satisfaction.  The  events  which  that  History  relates, 
the  speculations  and  controversies  which  are  there  de 
scribed,  and  discussions  of  the  same  kind,  far  more 
extensive,  which  are  there  omitted,  must  all  be  taken 
into  our  account  at  present,  as  the  prominent  and 
standard  examples  of  the  circumstances  which  attend 
the  progress  of  knowledge.  With  so  much  of  real  his 
torical  fact  before  us,  we  may  hope  to  avoid  such  views 
of  the  processes  of  the  human  mind  as  are  too  partial 
and  limited,  or  too  vague  and  loose,  or  too  abstract  and 
unsubstantial,  to  represent  fitly  the  real  forms  of  dis 
covery  and  of  truth. 

Of  former  attempts,  made  with  the  same  viewr  of 
tracing  the  conditions  of  the  progress  of  knowledge,  that 
of  Bacon  is  perhaps  the  most  conspicuous :  and  his 
labours  on  this  subject  were  opened  by  his  book  on  the 
Advancement  of  Learning,  which  contains,  among  other 
matter,  a  survey  of  the  then  existing  state  of  knowledge. 
But  this  review  was  undertaken  rather  with  the  object 
of  ascertaining  in  what  quarters  future  advances  were  to 
be  hoped  for,  than  of  learning  by  what  means  they  were 
to  be  made.  His  examination  of  the  domain  of  human 
knowledge  was  conducted  rather  with  the  view  of  dis 
covering  what  remained  undone,  than  of  finding  out  how 
so  much  had  been  done.  Bacon's  survey  was  made  for 
the  purpose  of  tracing  the  boundaries,  rather  than  of 
detecting  the  principles  of  knowledge.  "I  will  now 
attempt,"  he  says*,  "to  make  a  general  and  faithful 
perambulation  of  learning,  with  an  inquiry  what  parts 
thereof  lie  fresh  and  waste,  and  not  improved  and  con 
verted  by  the  industry  of  man ;  to  the  end  that  such  a 
plot  made  and  recorded  to  memory,  may  both  minister 

*  Advancement  of  Learning,  1>.  i.  p.  74. 


light  to  any  public  designation,  and  also  serve  to  excite 
voluntary  endeavours."  Nor  will  it  be  foreign  to  our 
scheme  also  hereafter  to  examine  with  a  like  purpose 
the  frontier-line  of  man's  intellectual  estate.  But  the 
object  of  our  perambulation  in  the  first  place,  is  not  so 
much  to  determine  the  extent  of  the  field,  as  the  sources 
of  its  fertility.  We  would  learn  by  what  plan  and  rules 
of  culture,  conspiring  with  the  native  forces  of  the  boun 
teous  soil,  those  rich  harvests  have  been  produced  which 
fill  our  garners.  Bacon's  maxims,  on  the  other  hand, 
respecting  the  mode  in  which  he  conceived  that  know 
ledge  was  thenceforth  to  be  cultivated,  have  little  refer 
ence  to  the  failures,  still  less  to  the  successes,  which  are 
recorded  in  his  Review  of  the  learning  of  his  time.  His 
precepts  are  connected  with  his  historical  views  in  a 
slight  and  unessential  manner.  His  Philosophy  of  the 
Sciences  is  not  collected  from  the  Sciences  which  are 
noticed  in  his  survey.  Nor,  in  truth,  could  this,  at  the 
time  when  he  wrote,  have  easily  been  otherwise.  At 
that  period,  scarce  any  branch  of  physics  existed  as  a 
science,  except  Astronomy.  The  rules  which  Bacon  gives 
for  the  conduct  of  scientific  researches  are  obtained,  as 
it  were,  by  divination,  from  the  contemplation  of  sub 
jects  with  regard  to  which  no  sciences  as  yet  were.  His 
instances  of  steps  rightly  or  wrongly  made  in  this  path, 
are  in  a  great  measure  cases  of  his  own  devising.  He 
could  not  have  exemplified  his  Aphorisms  by  references 
to  treatises  then  extant,  on  the  laws  of  nature ;  for  the 
constant  burden  of  his  exhortation  is,  that  men  up  to 
his  time  had  almost  universally  followed  an  erroneous 
course.  And  however  we  may  admire  the  sagacity  with 
which  he  pointed  the  way  along  a  better  path,  we  have 
this  great  advantage  over  him ; — that  we  can  interrogate 
the  many  travellers  who  since  his  time  have  journeyed 
on  this  road.  At  the  present  day,  when  we  have  under 

12  OF    IDEAS    IX    GENERAL. 

our  notice  so  many  sciences,  of  such  wide  extent,  so  well 
established  ;  a  Philosophy  of  the  Sciences  ought,  it  must 
seem,  to  be  founded,  not  upon  conjecture,  but  upon  an 
examination  of  many  instances; — should  not  consist  of 
a  few  vague  and  unconnected  maxims,  difficult  and 
doubtful  in  their  application,  but  should  form  a  system 
of  which  every  part  has  been  repeatedly  confirmed  and 

This  accordingly  it  is  the  purpose  of  the  present 
work  to  attempt.  But  I  may  further  observe,  that  as 
my  hope  of  making  any  progress  in  this  undertaking  is 
founded  upon  the  design  of  keeping  constantly  in  view 
the  whole  result  of  the  past  history  and  present  con 
dition  of  science,  I  have  also  been  led  to  draw  my  les 
sons  from  my  examples  in  a  manner  more  systematic 
and  regular,  as  appears  to  me,  than  has  been  done  by 
preceding  writers.  Bacon,  as  I  have  just  said,  was  led 
to  his  maxims  for  the  promotion  of  knowledge  by  the 
sagacity  of  his  own  mind,  with  little  or  no  aid  from 
previous  examples.  Succeeding  philosophers  may  often 
have  gathered  useful  instruction  from  the  instances  of 
scientific  truths  and  discoveries  which  thev  adduced,  but 


their  conclusions  were  drawn  from  their  instances  casu 
ally  and  arbitrarily.  They  took  for  their  moral  any 
which  the  story  might  suggest.  But  such  a  proceeding 
as  this  cannot  suffice  for  us,  whose  aim  is  to  obtain  a 
consistent  body  of  philosophy  from  a  contemplation  of 
the  whole  of  Science  and  its  History.  For  our  purpose 
it  is  necessary  to  resolve  scientific  truths  into  their  con 
ditions  and  ingredients,  in  order  that  we  may  see  in 
what  manner  each  of  these  has  been  and  is  to  be  pro 
vided,  in  the  cases  which  we  may  have  to  consider.  This 
accordingly  is  necessarily  the  first  part  of  our  task  : — to 
analyze  Scientific  Truth  into  its  Elements.  This  attempt 
will  occupy  the  earlier  portion  of  the  present  work  ;  and 


will  necessarily  be  somewhat  long,  and  perhaps,  in  many 
parts,  abstruse  and  uninviting.  The  risk  of  such  an 
inconvenience  is  inevitable  ;  for  the  inquiry  brings  before 
us  many  of  the  most  dark  and  entangled  questions  in 
which  men  have  at  any  time  busied  themselves.  And 
even  if  these  can  now  be  made  clearer  and  plainer  than 
of  yore,  still  they  can  be  made  so  only  by  means  of  men 
tal  discipline  and  mental  effort.  Moreover  this  analysis 
of  scientific  truth  into  its  elements  contains  much,  both 
in  its  principles  and  in  its  results,  different  from  the 
doctrines  most  generally  prevalent  among  us  in  recent 
times :  but  on  that  very  account  this  analysis  is  an 
essential  part  of  the  doctrines  which  I  have  now  to  lay 
before  the  reader:  and  I  must  therefore  crave  his 
indulgence  towards  any  portion  of  it  which  may  appear 
to  him  obscure  or  repulsive. 

There  is  another  circumstance  which  may  tend  to 
make  the  present  work  less  pleasing  than  others  on  the 
same  subject,  in  the  nature  of  the  examples  of  human 
knowledge  to  wrhich  I  confine  myself;  all  my  instances 
being,  as  I  have  said,  taken  from  the  material  sciences. 
For  the  truths  belonging  to  these  sciences  are,  for  the 
most  part,  neither  so  familiar  nor  so  interesting  to  the 
bulk  of  readers  as  those  doctrines  which  belong  to  some 
other  subjects.  Every  general  proposition  concerning 
politics  or  morals  at  once  stirs  up  an  interest  in  men's 
bosoms,  which  makes  them  listen  with  curiosity  to  the 
attempts  to  trace  it  to  its  origin  and  foundation.  Every 
rule  of  art  or  language  brings  before  the  mind  of  culti 
vated  men  subjects  of  familiar  and  agreeable  thought, 
and  is  dwelt  upon  with  pleasure  for  its  own  sake,  as  well 
as  on  account  of  the  philosophical  lessons  which  it  may 
convey.  But  the  curiosity  which  regards  the  truths  of 
physics  or  chemistry,  or  even  of  physiology  and  astro 
nomy,  is  of  a  more  limited  and  less  animated  kind. 
•< ' 

14  OF    IDEAS    IN    GENERAL. 

Hence,  in  the  mode  of  inquiry  which  I  have  prescribed 
to  myself,  the  examples  which  I  have  to  adduce  will  not 
amuse  and  relieve  the  reader's  mind  as  much  as  they 
might  do,  if  I  could  allow  myself  to  collect  them  from 
the  whole  field  of  human  knowledge.  They  will  have  in 
them  nothing  to  engage  his  fancy,  or  to  warm  his  heart. 
I  am  compelled  to  detain  the  listener  in  the  chilly  air 
of  the  external  world,  in  order  that  we  may  have  the 
advantage  of  full  daylight. 

But  although  I  cannot  avoid  this  inconvenience,  so 
far  as  it  is  one,  I  hope  it  will  be  recollected  how  great 
are  the  advantages  which  we  obtain  by  this  restriction. 
We  are  thus  enabled  to  draw  all  our  conclusions  from 
doctrines  which  are  universally  allowed  to  be  eminently 
certain,  clear,  and  definite.  The  portions  of  knowledge 
to  which  1  refer  are  well  known,  and  well  established 
among  men,  Their  names  are  familiar,  their  assertions 
uncontested.  Astronomy  and  Geology,  Mechanics  and 
Chemistry,  Optics  and  Acoustics,  Botany  and  Physiology, 
are  each  recognized  as  large  and  substantial  collections 
of  undoubted  truths.  Men  are  wont  to  dwell  with  pride 
and  triumph  on  the  acquisitions  of  knowledge  which 
have  been  made  in  each  of  these  provinces ;  and  to  speak 
with  confidence  of  the  certainty  of  their  results.  And  all 
can  easily  learn  in  what  repositories  these  treasures  of 
human  knowledge  are  to  be  found.  When,  therefore, 
we  begin  our  inquiry  from  such  examples,  we  proceed 
upon  a  solid  foundation.  With  such  a  clear  ground  of 
confidence,  we  shall  not  be  met  with  general  assertions 
of  the  vagueness  and  uncertainty  of  human  knowledge  ; 
with  the  question,  What  truth  is,  and  How  we  are  to 
recognize  it ;  with  complaints  concerning  the  hopeless 
ness  and  unprofitableness  of  such  researches.  We  have, 
at  least,  a  definite  problem  before  us.  We  have  to 
examine  the  structure  and  scheme,  not  of  a  shapeless 


mass  of  incoherent  materials,  of  which  we  doubt  whether 
it  be  a  ruin  or  a  natural  wilderness,  but  of  a  fair  and 
lofty  palace,  still  erect  and  tenanted,  where  hundreds  of 
different  apartments  belong  to  a  common  plan,  where 
every  generation  adds  something  to  the  extent  and  mag 
nificence  of  the  pile.  The  certainty  and  the  constant 
progress  of  science  are  things  so  unquestioned,  that  we 
are  at  least  engaged  in  an  intelligible  inquiry,  when  we 
are  examining  the  grounds  and  nature  of  that  certainty, 
the  causes  and  laws  of  that  progress. 

To  this  enquiry,  then,  we  now  proceed.  And  in 
entering  upon  this  task,  however  our  plan  or  our  prin 
ciples  may  differ  from  those  of  the  eminent  philosophers 
who  have  endeavoured,  in  our  own  or  in  former  times, 
to  illustrate  or  enforce  the  philosophy  of  science,  we 
most  willingly  acknowledge  them  as  in  many  things  our 
leaders  and  teachers.  Each  reform  must  involve  its  own 
peculiar  principles,  and  the  result  of  our  attempts,  so 
far  as  they  lead  to  a  result,  must  be,  in  some  respects, 
different  from  those  of  former  works.  But  we  may  still 
share  with  the  great  writers  who  have  treated  this 
subject  before  us,  their  spirit  of  hope  and  trust,  their 
reverence  for  the  dignity  of  the  subject,  their  belief  in 
the  vast  powers  and  boundless  destiny  of  man.  And  we 
may  once  more  venture  to  use  the  words  of  hopeful 
exhortation,  with  which  the  greatest  of  those  who  have 
trodden  this  path  encouraged  himself  and  his  followers 
when  he  set  out  upon  his  way. 

"  Concerning  ourselves  we  speak  not ;  but  as  touch 
ing  the  matter  which  we  have  in  hand,  this  we  ask;— 
that  men  deem  it  not  to  be  the  setting  up  an  Opinion, 
but  the  performing  of  a  Work :  and  that  they  receive 
this  as  a  certainty ;  that  we  are  not  laying  the  founda 
tions  of  any  sect  or  doctrine,  but  of  the  profit  and 
dignity  of  mankind.  Furthermore,  that  being  well  dis- 

10  OF    IDEAS    IN    GENERAL. 

posed  to  what  shall  advantage  themselves,  and  putting 
off  factions  and  prejudices,  they  take  common  counsel 
with  us,  to  the  end  that  being  by  these  our  aids  and 
appliances  freed  and  defended  from  wanderings  and 
impediments,  they  may  lend  their  hands  also  to  the 
labours  which  remain  to  be  performed :  and  yet  further, 
that  they  be  of  good  hope ;  neither  imagine  to  them 
selves  this  our  Reform  as  something  of  infinite  dimen 
sion,  and  beyond  the  grasp  of  mortal  man,  when  in  truth 
it  is  the  end  and  true  limit  of  infinite  errour ;  and  is  by 
no  means  unmindful  of  the  condition  of  mortality  and 
humanity,  not  confiding  that  such  a  thing  can  be  carried 
to  its  perfect  close  in  the  space  of  one  single  age,  but 
assigning  it  as  a  task  to  a  succession  of  generations." 



SECT.  1. — Thoughts  and  Things. 

IN  order  that  we  may  do  something  towards  determining 
the  nature  and  conditions  of  human  knowledge,  (which 
I  have  already  stated  as  the  purpose  of  this  work,)  I 
shall  have  to  refer  to  an  antithesis  or  opposition,  which 
is  familiar  and  generally  recognized,  and  in  which  the 
distinction  of  the  things  opposed  to  each  other  is  com 
monly  considered  very  clear  and  plain.  I  shall  have  to 
attempt  to  make  this  opposition  sharper  and  stronger 
than  it  is  usually  conceived,  and  yet  to  shew  that  the 
distinction  is  far  from  being  so  clear  and  definite  as  it  is 
usually  assumed  to  be  :  I  shall  have  to  point  the  con 
trast,  yet  shew  that  the  things  which  are  contrasted 


cannot  be  separated : — I  must  explain  that  the  anti 
thesis  is  constant  and  essential,  but  yet  that  there  is  no 
fixed  and  permanent  line  dividing  its  members.  I  may 
thus  appear,  in  different  parts  of  my  discussion,  to  be 
proceeding  in  opposite  directions,  but  I  hope  that  the 
reader  who  gives  me  a  patient  attention  will  see  that 
both  steps  lead  to  the  point  of  view  to  which  I  wish  to 
lead  him. 

The  antithesis  or  opposition  of  which  I  speak  is 
denoted,  with  various  modifications,  by  various  pairs  of 
terms :  I  shall  endeavour  to  show  the  connexion  of  these 
different  modes  of  expression,  and  I  will  begin  with  that 
form  which  is  the  simplest  and  most  idiomatic. 

The  simplest  and  most  idiomatic  expression  of  the 
antithesis  to  which  I  refer  is  that  in  which  we  oppose  to 
each  other  THINGS  and  THOUGHTS.  The  opposition  is 
familiar  and  plain.  Our  Thoughts  are  something  which 
belongs  to  ourselves ;  something  which  takes  place 
within  us;  they  are  what  me  think ;  they  are  actions  of 
our  minds.  Things,  on  the  contrary,  are  something 
different  from  ourselves  and  independent  of  us ;  some 
thing  which  is  without  us ;  they  are ;  we  see  them, 
touch  them,  and  thus  know  that  they  exist ;  but  we  do 
not  make  them  by  seeing  or  touching  them,  as  we  make 
our  Thoughts  by  thinking  them ;  we  are  passive,  and 
Things  act  upon  our  organs  of  perception. 

Now  what  I  wish  especially  to  remark  is  this :  that 
in  all  human  KNOWLEDGE  both  Thoughts  and  Things  are 
concerned.  In  every  part  of  my  knowledge  there  must 
i)e  some  thing  about  which  I  know,  and  an  internal  act 
of  me  who  know.  Thus,  to  take  simple  yet  definite  parts 
of  our  knowledge,  if  I  know  that  a  solar  year  consists  of 
365  days,  or  a  lunar  month  of  30  days,  I  know'  some 
thing  about  the  sun  or  the  moon ;  namely,  that  those 
objects  perform  certain  revolutions  and  go  through  cer- 
VOL.  i.  \v.  p.  C 

18  OF    IDEAS   IN    GENERAL. 

tain  changes,  in  those  numbers  of  days;  but  I  count 
such  numbers  and  conceive  such  revolutions  and  changes 
by  acts  of  my  own  thoughts.  And  both  these  elements 
of  my  knowledge  are  indispensable.  If  there  were  not 
such  external  Things  as  the  sun  and  the  moon  I  could 
not  have  any  knowledge  of  the  progress  of  time  as 
marked  by  them.  And  however  regular  were  the  mo 
tions  of  the  sun  and  moon,  if  I  could  not  count  their 
appearances  and  combine  their  changes  into  a  cycle,  or 
if  I  could  not  understand  this  when  done  by  other  men, 
I  could  not  know  anything  about  a  year  or  a  month.  In 
the  former  case  I  might  be  conceived  as  a  human  being, 
possessing  the  human  powers  of  thinking  and  reckoning, 
but  kept  in  a  dark  world  with  nothing  to  mark  the  pro 
gress  of  existence,  The  latter  is  the  case  of  brute  ani 
mals,  which  see  the  sun  and  moon,  but  do  not  know  how 
many  days  make  a  month  or  a  year,  because  they  have 
not  human  powers  of  thinking  and  reckoning. 

The  two  elements  which  are  essential  to  our  know 
ledge  in  the  above  cases,  are  necessary  to  human  know 
ledge  in  all  cases.  In  all  cases,  Knowledge  implies  a 
combination  of  Thoughts  and  Things.  Without  this 
combination,  it  would  not  be  Knowledge.  Without 
Thoughts,  there  could  be  no  connexion ;  without  Things, 
there  could  be  no  reality.  Thoughts  and  Things  are  so 
intimately  combined  in  our  Knowledge,  that  we  do  not 
look  upon  them  as  distinct.  One  single  act  of  the  mind 
involves  them  both ;  and  their  contrast  disappears  in 
their  union. 

But  though  Knowledge  requires  the  union  of  these 
two  elements,  Philosophy  requires  the  separation  of 
them,  in  order  that  the  nature  and  structure  of  Know 
ledge  may  be  seen.  Therefore  I  begin  by  considering 
this  separation.  And  I  now  proceed  to  speak  of  another 
way  of  looking  at  the  antithesis  of  which  I  have  spoken; 


and  which  I  may,  for  the  reasons  which  I  have  just 
mentioned,  call  the  FUNDAMENTAL  ANTITHESIS  OF  PHI 

SECT.  2. — Necessary  and  Experiential  Truths. 

MOST  persons  are  familiar  with  the  distinction  of  ne 
cessary  and  contingent  truths.  The  former  kind  are 
Truths  which  cannot  but  be  true;  as  that  19  and  11 
make  30  ; — that  parallelograms  upon  the  same  base  and 
between  the  same  parallels  are  equal: — that  all  the 
angles  in  the  same  segment  of  a  circle  are  equal.  The 
latter  are  Truths  which  it  happens  (contingit]  are  true ; 
but  which,  for  any  thing  which  we  can  see,  might  have 
been  otherwise ;  as  that  a  lunar  month  contains  30  days, 
or  that  the  stars  revolve  in  circles  round  the  pole.  The 
latter  kind  of  Truths  are  learnt  by  experience,  and  hence 
we  may  call  them  Truths  of  Experience,  or,  for  the  sake 
of  convenience,  Experiential  Truths,  in  contrast  with 
Necessary  Truths. 

Geometrical  propositions  are  the  most  manifest  ex 
amples  of  Necessary  Truths.  All  persons  who  have  read 
and  understood  the  elements  of  geometry,  know  that  the 
propositions  above  stated  (that  parallelograms  upon  the 
same  base  and  between  the  same  parallels  are  equal ; 
that  all  the  angles  in  the  same  segment  of  a  circle  are 
equal,)  are  necessarily  true ;  not  only  they  are  true,  but 
they  must  le  true.  The  meaning  of  the  terms  being 
understood,  and  the  proof  being  gone  through,  the  truth 
of  the  propositions  must  be  assented  to.  We  learn  these 
propositions  to  be  true  by  demonstrations  deduced  from 
definitions  and  axioms ;  and  when  we  have  thus  learnt 
them,  we  see  that  they  could  not  be  otherwise.  In  the 
same  manner,  the  truths  which  concern  numbers  are 
necessary  truths:  19  and  11  not  only  do  make  30,  but 
must  make  that  number,  and  cannot  make  anything  else. 

20  OF    IDEAS  IN    GENERAL. 

In  the  same  manner,  it  is  a  necessary  truth  that  half  the 
sum  of  two  numbers  added  to  half  their  difference  is 
equal  to  the  greater  number. 

It  is  easy  to  find  examples  of  Experiential  Truths  ;— 
propositions  which  we  know  to  be  true,  but  know  by 
experience  only.  We  know,  in  this  way,  that  salt  will 
dissolve  in  water ;  that  plants  cannot  live  without  light ; 
— in  short,  we  know  in  this  way  all  that  we  do  know 
in  chemistry,  physiology,  and  the  material  sciences  in 
general.  I  take  the  Sciences  as  my  examples  of  human 
knowledge,  rather  than  the  common  truths  of  daily  life, 
or  moral  or  political  truths ;  because,  though  the  latter 
are  more  generally  interesting,  the  former  are  much 
more  definite  and  certain,  and  therefore  better  starting- 
points  for  our  speculations,  as  I  have  already  said.  And 
we  may  take  elementary  astronomical  truths  as  the  most 
familiar  examples  of  Experiential  Truths  in  the  domain 
of  science. 

With  these  examples,  the  distinction  of  Necessary 
and  Experiential  Truths  is,  I  hope,  clear.  The  former 
kind,  we  see  to  be  true  by  thinking  about  them,  and  see 
that  they  could  not  be  otherwise.  The  latter  kind,  men 
could  never  have  discovered  to  be  true  without  looking 
at  them ;  and  having  so  discovered  them,  still  no  one  will 
pretend  to  say  they  might  not  have  been  otherwise.  For 
aught  we  can  see,  the  astronomical  truths  which  express 
the  motions  and  periods  of  the  sun,  moon  and  stars, 
might  have  been  otherwise.  If  we  had  been  placed  in 
another  part  of  the  solar  system,  our  experiential  truths 
respecting  days,  years,  and  the  motions  of  the  heavenly 
bodies,  would  have  been  other  than  they  are,  as  we 
know  from  astronomy  itself. 

It  is  evident  that  this  distinction  of  Necessary  and 
Experiential  Truths  involves  the  same  antithesis  which 
we  have  already  considered ; — the  antithesis  of  Thoughts 


and  Things.  Necessary  Truths  are  derived  from  our  own 
Thoughts :  Experiential  Truths  are  derived  from  our 
observation  of  Things  about  us.  The  opposition  of 
Necessary  and  Experiential  Truths  is  another  aspect  of 
the  Fundamental  Antithesis  of  Philosophy. 

SECT.  3. — Deduction  and  Induction. 

I  HAVE  already  stated  that  geometrical  truths  are 
established  by  demonstrations  deduced  from  definitions 
and  axioms.  The  term  Deduction  is  specially  applied 
to  such  a  course  of  demonstration  of  truths  from  defini 
tions  and  axioms.  In  the  case  of  the  parallelograms 
upon  the  same  base  and  between  the  same  parallels,  we 
prove  certain  triangles  to  be  equal,  by  supposing  them 
placed  so  that  their  two  bases  have  the  same  extremi 
ties;  and  hence,  referring  to  an  Axiom  respecting  straight 
lines,  we  infer  that  the  bases  coincide.  We  combine 
these  equal  triangles  with  other  equal  spaces,  and  in  this 
way  make  up  both  the  one  and  the  other  of  the  paral 
lelograms,  in  such  a  manner  as  to  shew  that  they  are 
equal.  In  this  manner,  going  on  step  by  step,  deducing 
the  equality  of  the  triangles  from  the  axiom,  and  the 
equality  of  the  parallelograms  from  that  of  the  triangles, 
we  travel  to  the  conclusion.  And  this  process  of  suc 
cessive  deduction  is  the  scheme  of  all  geometrical  proof. 
We  begin  with  Definitions  of  the  notions  which  we  reason 
about,  and  with  Axioms,  or  self-evident  truths,  respecting 
these  notions;  and  we  get,  by  reasoning  from  these,  other 
truths  which  are  demonstratively  evident ;  and  from 
these  truths  again,  others  of  the  same  kind,  and  so  on. 
We  begin  with  our  own  Thoughts,  which  supply  us  with 
Axioms  to  start  from;  and  we  reason  from  these,  till  we 
rome  to  propositions  which  are  applicable  to  the  Things 
about  us;  as  for  instance,  the  propositions  respecting 
circles  and  spheres  are  applicable  to  the  motions  of  the 

22  OF    IDEAS   IN    GENERAL. 

heavenly  bodies.  This  is  Deduction,  or  Deductive  Rea 

Experiential  truths  are  acquired  in  a  very  different 
way.  In  order  to  obtain  such  truths,  we  begin  with 
Things.  In  order  to  learn  how  many  days  there  are  in 
a  year,  or  in  a  lunar  month,  we  must  begin  by  observing 
the  sun  and  the  moon.  We  must  observe  their  changes 
day  by  day,  and  try  to  make  the  cycle  of  change  fit  into 
some  notion  of  number  which  we  supply  from  our  own 
Thoughts.  We  shall  find  that  a  cycle  of  30  days  nearly 
will  fit  the  changes  of  phase  of  the  moon; — that  a  cycle 
of  365  days  nearly  will  fit  the  changes  of  daily  motion 
of  the  sun.  Or,  to  go  on  to  experiential  truths  of 
which  the  discovery  comes  within  the  limits  of  the  his 
tory  of  science — we  shall  find  (as  Hipparchus  found) 
that  the  unequal  motion  of  the  sun  among  the  stars, 
such  as  observation  shews  it  to  be,  may  be  fitly  repre 
sented  by  the  notion  of  an  eccentric; — a  circle  in  which 
the  sun  has  an  equable  annual  motion,  the  spectator  not 
being  in  the  center  of  the  circle.  Again,  in  the  same 
manner,  at  a  later  period,  Kepler  started  from  more 
exact  observations  of  the  sun,  and  compared  them  with 
a  supposed  motion  in  a  certain  ellipse;  and  was  able  to 
shew  that,  not  a  circle  about  an  eccentric  point,  but  an 
ellipse,  supplied  the  mode  of  conception  which  truly 
agreed  with  the  motion  of  the  sun  about  the  earth ;  or 
rather,  as  Copernicus  had  already  shewn,  of  the  earth 
about  the  sun.  In  such  cases,  in  which  truths  are  ob 
tained  by  beginning  from  observation  of  external  things 
and  by  finding  some  notion  with  which  the  Things,  as 
observed,  agree,  the  truths  are  said  to  be  obtained  by 
Induction.  The  process  is  an  Inductive  Process. 

The  contrast  of  the  Deductive  and  Inductive  process 
is  obvious.  In  the  former,  we  proceed  at  each  step 
from  general  truths  to  particular  applications  of  them ; 


in  the  latter,  from  particular  observations  to  a  general 
truth  which  includes  them.  In  the  former  case  we 
may  be  said  to  reason  downwards,  in  the  latter  case, 
upwards;  for  general  notions  are  conceived  as  stand 
ing  above  particulars.  Necessary  truths  are  proved, 
like  arithmetical  sums,  by  adding  together  the  portions 
of  which  they  consist.  An  inductive  truth  is  proved, 
like  the  guess  which  answers  a  riddle,  by  its  agreeing 
with  the  facts  described.  Dcmonstation  is  irresistible 
in  its  effect  on  the  belief,  but  does  not  produce  surprize, 
because  all  the  steps  to  the  conclusion  are  exhibited, 
before  we  arrive  at  the  conclusion.  Inductive  infer 
ence  is  not  demonstrative,  but  it  is  often  more  striking 
than  demonstrative  reasoning,  because  the  intermediate 
links  between  the  particulars  and  the  inference  are  not 
shown.  Deductive  truths  are  the  results  of  relations 
among  our  own  Thoughts.  Inductive  Truths  are  re 
lations  which  we  discern  among  existing  Things;  and 
thus,  this  opposition  of  Deduction  and  Induction  is  again 
an  aspect  of  the  Fundamental  Antithesis  already  spoken 

SECT.  4. — Theories  and  Facts. 

GENERAL  experiential  Truths,  such  as  we  have  just 
spoken  of,  are  called  Theories,  and  the  particular 
observations  from  which  they  are  collected,  and  which 
they  include  and  explain,  are  called  Facts.  Thus  Hip- 
parchus's  doctrine,  that  the  sun  moves  in  an  eccentric 
about  the  earth,  is  his  Theory  of  the  Sun,  or  the  Eccen 
tric  Theory.  The  doctrine  of  Kepler,  that  the  Earth 
moves  in  an  Ellipse  about  the  Sun,  is  Kepler  s  Theory 
of  the  Earth,  the  Elliptical  Theory.  Newton's  doctrine 
that  this  elliptical  motion  of  the  Earth  about  the  Sun 
is  produced  and  governed  by  the  Sun's  attraction  upon 
the  Earth,  is  the  Newtonian  theory,  the  Theory  of 
Attraction.  Each  of  these  Theories  was  accepted,  be- 


cause  it  included,  connected  and  explained  the  Facts; 
the  Facts  being,  in  the  two  former  cases,  the  motions 
of  the  Sun  as  observed;  and  in  the  other  case,  the  ellip 
tical  motion  of  the  Earth  as  known  by  Kepler's  Theory. 
This  antithesis  of  Theory  and  Fact  is  included  in  what 
has  just  been  said  of  Inductive  Propositions.  A  Theory 
is  an  Inductive  Proposition,  and  the  Facts  are  the  par 
ticular  observations  from  which,  as  I  have  said,  such 
Propositions  are  inferred  by  Induction.  The  Antithesis 
of  Theory  and  Fact  implies  the  fundamental  Antithesis 
of  Thoughts  and  Things;, for  a  Theory  (that  is,  a  true 
Theory)  may  be  described  as  a  Thought  which  is  con 
templated  distinct  from  Things  and  seen  to  agree  with 
them;  while  a  Fact  is  a  combination  of  our  Thoughts 
with  Things  in  so  complete  agreement  that  we  do  not 
regard  them  as  separate. 

Thus  the  antithesis  of  Theory  and  Fact  involves  the 
antithesis  of  Thoughts  and  Things,  but  is  not  identical 
with  it.  Facts  involve  Thoughts,  for  we  know  Facts  only 
by  thinking  about  them.  The  Fact  that  the  year  consists 
of  365  days;  the  Fact  that  the  month  consists  of  30  days, 
cannot  be  known  to  us,  except  we  have  the  Thoughts 
of  Time,  Number  and  Recurrence.  But  these  Thoughts 
are  so  familiar,  that  we  have  the  Fact  in  our  mind 
as  a  simple  Thing  without  attending  to  the  Thought 
which  it  involves.  When  we  mould  our  Thoughts  into  a 
Theory,  we  consider  the  Thought  as  distinct  from  the 
Facts;  but  yet,  though  distinct,  not  independent  of  them; 
for  it  is  a  true  Theory,  only  by  including  and  agreeing 
with  the  Facts. 

SECT.  5. — Ideas  and  Sensations. 

WE  have  just  seen  that  the  antithesis  of  Theory  and 
Fact,  although  it  involves  the  antithesis  of  Thoughts  and 
Things,  is  not  identical  with  it.  There  are  other  modes 


of  expression  also,  which  involve  the  same  Fundamental 
Antithesis,  more  or  less  modified.  Of  these,  the  pair  of 
words  which  in  their  relations  appear  to  separate  the 
members  of  the  antithesis  most  distinctly  are  Ideas  and 
Sensations.  We  see  and  hear  and  touch  external  things, 
and  thus  perceive  them  by  our  senses;  but  in  perceiving 
them,  we  connect  the  impressions  of  sense  according  to 
relations  of  space,  time,  number,  likeness,  cause,  &c. 
Now  some  at  least  of  these  kinds  of  connexion,  as  space, 
time,  number,  may  be  contemplated  distinct  from  the 
things  to  which  they  are  applied;  and  so  contemplated, 
I  term  them  Ideas.  And  the  other  element,  the  impres 
sions  upon  our  senses  which  they  connect,  are  called 

I  term  space,  time,  cause,  &c.,  Ideas,  because  they 
are  general  relations  among  our  sensations,  apprehend 
ed  by  an  act  of  the  mind,  not  by  the  senses  simply. 
These  relations  involve  something  beyond  what  the 
senses  alone  could  furnish.  By  the  sense  of  sight  we 
see  various  shades  and  colours  and  shapes  before  us,  but 
the  outlines  by  which  they  are  separated  into  distinct 
objects  of  definite  forms,  are  the  work  of  the  mind  itself. 
And  again,  when  we  conceive  visible  things,  not  only  as 
surfaces  of  a  certain  form,  but  as  solid  bodies,  placed  at 
various  distances  in  space,  we  again  exert  an  act  of  the 
mind  upon  them.  When  we  see  a  body  move,  we  see 
it  move  in  a  path  or  orbit,  but  this  orbit  is  not  itself 
seen;  it  is  constructed  by  the  mind.  In  like  manner 
when  we  see  the  motions  of  a  needle  towards  a  mag 
net,  we  do  not  see  the  attraction  or  force  which  pro 
duces  the  effects;  but  we  infer  the  force,  by  having  in 
our  minds  the  Idea  of  Cause.  Such  acts  of  thought, 
such  Ideas,  enter  into  our  perceptions  of  external  things. 

But  though  our  perceptions  of  external  things  in 
volve  some  act  of  the  mind,  they  must  involve  some- 


thing  else  besides  an  act  of  the  mind.  If  we  must  exer 
cise  an  act  of  thought  in  order  to  see  force  exerted,  or 
orbits  described  by  bodies  in  motion,  or  even  in  order 
to  see  bodies  existing  in  space,  and  to  distinguish  one 
kind  of  object  from  another,  still  the  act  of  thought 
alone  does  not  make  the  bodies.  There  must  be  some 
thing  besides,  on  mldcli  the  thought  is  exerted.  A 
colour,  a  form,  a  sound,  are  not  produced  by  the  mind, 
however  they  may  be  moulded,  combined,  and  inter 
preted  by  our  mental  acts.  A  philosophical  poet  has 
spoken  of 

All  the  world 

Of  eye  and  ear,  both  what  they  half  create, 
And  what  perceive. 

But  it  is  clear,  that  though  they  half  create,  they  do  not 
wholly  create:  there  must  be  an  external  world  of  colour 
and  sound  to  give  impressions  to  the  eye  and  ear,  as 
well  as  internal  powers  by  which  we  perceive  what  is 
offered  to  our  organs.  The  mind  is  in  some  way  passive 
as  well  as  active:  there  are  objects  without  as  well  as 
faculties  within; — Sensations,  as  well  as  acts  of  Thought. 
Indeed  this  is  so  far  generally  acknowledged,  that 
according  to  common  apprehension,  the  mind  is  passive 
rather  than  active  in  acquiring  the  knowledge  which 
it  receives  concerning  the  material  world.  Its  sensa 
tions  are  generally  considered  more  distinct  than  its 
operations.  The  world  without  is  held  to  be  more  clearly 
real  than  the  faculties  within.  That  there  is  some 
thing  different  from  ourselves,  something  external  to  us, 
something  independent  of  us,  something  which  no  act 
of  our  minds  can  make  or  can  destroy,  is  held  by  all 
men  to  be  at  least  as  evident,  as  that  our  minds  can 
exert  any  effectual  process  in  modifying  and  appreciating 
the  impressions  made  upon  them.  Most  persons  are 
more  likely  to  doubt  whether  the  mind  be  always  actively 


applying  Ideas  to  the  objects  which  it  perceives,  than 
whether  it  perceive  them  passively  by  means  of  Sen 

But  yet  a  little  consideration  will  show  us  that  an 
activity  of  the  mind,  and  an  activity  according  to  certain 
Ideas,  is  requisite  in  all  our  knowledge  of  external 
objects.  We  see  objects,  of  various  solid  forms,  and  at 
various  distances  from  us.  But  we  do  not  thus  perceive 
them  by  sensation  alone.  Our  visual  impressions  can 
not,  of  themselves,  convey  to  us  a  knowledge  of  solid 
form,  or  of  distance  from  us.  Such  knowledge  is  inferred 
from  what  we  see : — inferred  by  conceiving  the  objects 
as  existing  in  space,  and  by  applying  to  them  the  Idea  of 
Space.  Again : — day  after  day  passes,  till  they  make  up  a 
year :  but  we  do  not  know  that  the  days  are  365,  except 
we  count  them;  and  thus  apply  to  them  our  Idea  of  Num 
ber.  Again  : — we  see  a  needle  drawn  to  a  magnet :  but, 
in  truth,  the  drawing  is  what  we  cannot  see.  We  see  the 
needle  move,  and  infer  the  attraction,  by  applying  to  the 
fact  our  Idea  of  Force,  as  the  cause  of  motion.  Again: — 
we  see  two  trees  of  different  kinds ;  but  we  cannot  know 
that  they  are  so,  except  by  applying  to  them  our  Idea 
of  the  resemblance  and  difference  which  makes  kinds. 
And  thus  Ideas,  as  well  as  Sensations,  necessarily  enter 
into  all  our  knowledge  of  objects :  and  these  two  words 
express,  perhaps  more  exactly  than  any  of  the  pairs 
before  mentioned,  that  Fundamental  Antithesis,  in  the 
union  of  which,  as  I  have  said,  all  knowledge  consists. 

SECT  6. — Reflexion  and  Sensation. 

IT  will  hereafter  be  my  business  to  show  what  the 
Ideas  are,  which  thus  enter  into  our  knowledge ;  and 
how  each  Idea  has  been,  as  a  matter  of  historical  fact, 
introduced  into  the  Science  to  which  it  especially  be 
longs.  But  before  I  proceed  to  do  this,  I  will  notice 

28  OF   IDEAS    IN    GENERAL. 

some  other  terms,  besides  the  phrases  already  noticed, 
which  have  a  reference,  more  or  less  direct,  to  the  Funda 
mental  Antithesis  of  Ideas  and  Sensations.  I  will  mention 
some  of  these,  in  order  that  if  they  should  come  under 
the  reader's  notice,  he  may  not  be  perplexed  as  to  their 
bearing  upon  the  view  here  presented  to  him. 

The  celebrated  doctrine  of  Locke,  that  all  our 
"Ideas,"  (that  is,  in  his  use  of  the  word,  all  our  objects 
of  thinking,)  come  from  Sensation  or  Reflexion,  will 
naturally  occur  to  the  reader  as  connected  with  the 
antithesis  of  which  I  have  been  speaking.  But  there  is 
a  great  difference  between  Locke's  account  of  Sensation 
and  Reflexion,  and  our  view  of  Sensation  and  Ideas.  He 
is  speaking  of  the  origin  of  our  knowledge ; — we,  of  its 
nature  and  composition.  He  is  content  to  say  that  all 
the  knowledge  which  we  do  not  receive  directly  by 
Sensation,  we  obtain  by  Reflex  Acts  of  the  mind,  which 
make  up  his  Reflexion.  But  we  hold  that  there  is  no 
Sensation  without  an  act  of  the  mind,  and  that  the 
mind's  activity  is  not  only  reflexly  exerted  upon  itself, 
but  directly  upon  objects,  so  as  to  perceive  in  them  con 
nexions  and  relations  which  are  not  Sensations.  He  is 
content  to  put  together,  under  the  name  of  Reflexion, 
everything  in  our  knowledge  which  is  not  Sensation :  we 
are  to  attempt  to  analyze  all  that  is  not  Sensation ;  not 
only  to  say  it  consists  of  Ideas,  but  to  point  out  what 
those  Ideas  are,  and  to  show  the  mode  in  which  each  of 
them  enters  into  our  knowledge.  His  purpose  was,  to 
prove  that  there  are  no  Ideas,  except  the  reflex  acts  of 
the  mind  :  our  endeavour  will  be  to  show  that  the  acts  of 
the  mind,  both  direct  and  reflex,  are  governed  by  certain 
Laws,  which  may  be  conveniently  termed  Ideas.  His 
procedure  was,  to  deny  that  any  knowledge  could  be 
derived  from  the  mind  alone :  our  course  will  be,  to 
show  that  in  every  part  of  our  most  certain  and  exact 


knowledge,  those  who  have  added  to  our  knowledge  in 
every  age  have  referred  to  principles  which  the  mind 
itself  supplies.  I  do  not  say  that  my  view  is  contrary  to 
his :  but  it  is  altogether  different  from  his.  If  I  grant 
that  all  our  knowledge  comes  from  Sensation  and  Re 
flexion,  still  my  task  then  is  only  begun ;  for  I  want 
further  to  determine,  in  each  science,  what  portion 
comes,  not  from  mere  Sensation,  but  from  those  Ideas 
by  the  aid  of  which  either  Sensation  or  Reflexion  can 
lead  to  Science. 

Locke's  use  of  the  word  "idea"  is,  as  the  reader  will 
perceive,  different  from  ours.  He  uses  the  word,  as  he 
says,  which  "  serves  best  to  stand  for  whatsoever  is  the 
object  of  the  understanding  when  a  man  thinks."  "  I 
have  used  it,"  he  adds,  "  to  express  whatever  is  meant  by 
phantasm,  notion,  species,  or  whatever  it  is  to  which  the 
mind  can  be  employed  about  in  thinking."  It  might  be 
shown  that  this  separation  of  the  mind  itself  from  the 
ideal  objects  about  which  it  is  employed  in  thinking,  may 
lead  to  very  erroneous  results.  But  it  may  suffice  to  ob 
serve  that  we  use  the  word  Ideas,  in  the  manner  already 
explained,  to  express  that  element,  supplied  by  the  mind 
itself,  which  must  be  combined  with  Sensation  in  order 
to  produce  knowledge.  For  us,  Ideas  are  not  Objects  of 
Thought,  but  rather  Laws  of  Thought.  Ideas  are  not 
synonymous  with  Notions;  they  are  Principles  which 
give  to  our  Notions  whatever  they  contain  of  truth.  But 
our  use  of  the  term  Idea  will  be  more  fully  explained 

SECT.  7 — Subjective  and  Objective. 

THE  Fundamental  Antithesis  of  Philosophy  of  which  I 
have  to  speak  has  been  brought  into  great  prominence 
in  the  writings  of  modern  German  philosophers,  and  has 
conspicuously  formed  the  basis  of  their  systems.  They 

30  OF    IDEAS   IN    GENERAL. 

have  indicated  this  antithesis  by  the  terms  subject  Ire  and 
objective.  According  to  the  technical  language  of  old 
writers,  a  thing  and  its  qualities  are  described  as  subject 
and  attributes ;  and  thus  a  man's  faculties  and  acts  are 
attributes  of  which  he  is  the  subject.  The  mind  is  the 
sulject  in  which  ideas  inhere..  Moreover,  the  man's 
faculties  and  acts  are  employed  upon  external  objects; 
and  from  objects  all  his  sensations  arise.  Hence  the 
part  of  a  man's  knowledge  which  belongs  to  his  own 
mind,  is  subjective:  that  which  flows  in  upon  him  from 
the  world  external  to  him,  is  objective.  And  as  in  man's 
contemplation  of  nature,  there  is  always  some  act  of 
thought  which  depends  upon  himself,  and  some  matter 
of  thought  which  is  independent  of  him,  there  is,  in  every 
part  of  his  knowledge,  a  subjective  and  an  objective 
element.  The  combination  of  the  two  elements,  the 
subjective  or  ideal,  and  the  objective  or  observed,  is 
necessary,  in  order  to  give  us  any  insight  into  the  laws  of 
nature.  But  different  persons,  according  to  their  mental 
habits  and  constitution,  may  be  inclined  to  dwell  by 
preference  upon  the  one  or  the  other  of  these  two 
elements.  It  may  perhaps  interest  the  reader  to  see 
this  difference  of  intellectual  character  illustrated  in  two 
eminent  men  of  genius  of  modern  times,  Gothe  and 

Gothe  himself  gives  us  the  account  to  which  I  refer, 
in  his  history  of  the  progress  of  his  speculations  con 
cerning  the  Metamorphosis  of  Plants;  a  mode  of  viewing 
their  structure  by  which  he  explained,  in  a  very  striking 
and  beautiful  manner,  the  relations  of  the  different  parts 
of  a  plant  to  each  other ;  as  has  been  narrated  in  the 
History  of  the  Inductive  Sciences.  Gothe  felt  a  delight 
in  the  passive  contemplation  of  nature,  unmingled  with 
the  desire  of  reasoning  and  theorizing ;  a  delight  such  as 
naturally  belongs  to  those  poets  who  merely  embody  the 


images  which  a  fertile  genius  suggests,  and  do  not  mix 
with  these  pictures,  judgments  and  reflexions  of  their 
own.  Schiller,  on  the  other  hand,  both  by  his  own 
strong  feeling  of  the  value  of  a  moral  purpose  in  poetry, 
and  by  his  adoption  of  a  system  of  metaphysics  in  which 
the  subjective  element  was  made  very  prominent,  was 
well  disposed  to  recognize  fully  the  authority  of  ideas 
over  external  impressions. 

Gothe  for  a  time  felt  a  degree  of  estrangement 
towards  Schiller,  arising  from  this  contrariety  in  their 
views  and  characters.  But  on  one  occasion  they  fell 
into  discussion  on  the  study  of  natural  history;  and 
Gothe  endeavoured  to  impress  upon  his  companion  his 
persuasion  that  nature  was  to  be  considered,  not  as  com 
posed  of  detached  and  incoherent  parts,  but  as  active 
and  alive,  and  unfolding  herself  in  each  portion,  in 
virtue  of  principles  which  pervade  the  whole.  Schiller 
objected  that  no  such  view  of  the  objects  of  natural 
history  had  been  pointed  out  by  observation,  the  only 
guide  which  the  natural  historians  recommended;  and 
was  disposed  on  this  account  to  think  the  whole  of  their 
study  narrow  and  shallow.  "Upon  this,"  says  Gothe, 
"  I  expounded  to  him,  in  as  lively  a  way  as  I  could,  tho 
metamorphosis  of  plants,  drawing  on  paper  for  him,  as  I 
proceeded,  a  diagram  to  represent  that  general  form  of 
a  plant  which  shows  itself  in  so  many  and  so  various 
transformations.  Schiller  attended  and  understood;  and, 
accepting  the  explanation,  he  said,  '  This  is  not  observa 
tion,  but  an  idea.'  I  replied,"  adds  Gothe,  "with  some 
degree  of  irritation ;  for  the  point  which  separated  us 
was  most  luminously  marked  by  this  expression :  but  I 
smothered  my  vexation,  and  merely  said,  '  I  was  happy 
to  find  that  I  had  got  ideas  without  knowing  it ;  nay, 
that  I  saw  them  before  my  eyes.' "  Gothe  then  goes  on 
to  say,  that  he  had  been  grieved  to  the  very  soul  by 

32  OF    IDEAS    IX    GENERAL. 

maxims  promulgated  by  Schiller,  that  no  observed  fact 
ever  could  correspond  with  an  idea.  Since  he  himself 
loved  best  to  wander  in  the  domain  of  external  observa 
tion,  he  had  been  led  to  look  with  repugnance  and 
hostility  upon  anything  Avhich  professed  to  depend  upon 
ideas.  "Yet,"  he  observes,  "it  occurred  to  me  that  if 
my  Observation  was  identical  with  his  Idea,  there  must 
be  some  common  ground  on  which  we  might  meet." 
They  went  on  with  their  mutual  explanations,  and  be 
came  intimate  and  lasting  friends.  "And  thus,"  adds 
the  poet,  "  by  means  of  that  mighty  and  interminable 
controversy  between  object  and  subject,  we  two  concluded 
an  alliance  which  remained  unbroken,  and  produced 
much  benefit  to  ourselves  and  others." 

The  general  diagram  of  a  plant,  of  which  Gothe 
here  speaks,  must  have  been  a  combination  of  lines  and 
marks  expressing  the  relations  of  position  and  equiva 
lence  among  the  elements  of  vegetable  forms,  by  which 
so  many  of  their  resemblances  and  differences  may  be 
explained.  Such  a  symbol  is  not  an  Idea  in  that  general 
sense  in  which  we  propose  to  use  the  term,  but  is  a 
particular  modification  of  the  general  Ideas  of  symmetry, 
developement,  and  the  like  ;  and  we  shall  hereafter  see, 
according  to  the  phraseology  which  we  shall  explain  in 
the  next  chapter,  how  such  a  diagram  might  express 
the  ideal  conception  of  a  plant. 

The  antithesis  of  subjective  and  objective  is  very 
familiar  in  the  philosophical  literature  of  Germany  and 
France;  nor  is  it  uncommon  in  any  age  of  our  own 
literature.  But  though  efforts  have  recently  been  made 
to  give  currency  among  us  to  this  phraseology,  it  has 
not  been  cordially  received,  and  has  been  much  com 
plained  of  as  not  of  obvious  meaning.  ]STor  is  the  com 
plaint  without  ground :  for  when  we  regard  the  mind  as 
the  subject  in  which  ideas  inhere,  it  becomes  for  us  an 


object,  and  the  antithesis  vanishes.  We  are  not  so 
much  accustomed  to  use  subject  in  this  sense,  as  to 
make  it  a  proper  contrast  to  object.  The  combination 
" ideal  and  objective"  would  more  readily  convey  to  a 
modern  reader  the  opposition  which  is  intended  between 
the  ideas  of  the  mind  itself,  and  the  objects  which  it 
contemplates  around  it. 

To  the  antitheses  already  noticed — Thoughts  and 
Things ;  Necessary  and  Experiential  Truths ;  Deduction 
and  Induction ;  Theory  and  Fact ;  Ideas  and  Sensations ; 
Reflexion  and  Sensation ;  Subjective  and  Objective ;  we 
may  add  others,  by  which  distinctions  depending  more 
or  less  upon  the  fundamental  antithesis  have  been  de 
noted.  Thus  we  speak  of  the  internal  and  external 
sources  of  our  knowledge ;  of  the  world  within  and  the 
world  without  us ;  of  Man  and  Nature.  Some  of  the 
more  recent  metaphysical  writers  of  Germany  have 
divided  the  universe  into  the  Me  and  the  Not-me  (Ich 
and  Xicht-ich).  Upon  such  phraseology  we  may  observe, 
that  to  have  the  fundamental  antithesis  of  which  we 
speak  really  understood,  is  of  the  highest  consequence 
to  philosophy,  but  that  little  appears  to  be  gained  by 
expressing  it  in  any  novel  manner.  The  most  weighty 
part  of  the  philosopher's  task  is  to  analyze  the  operations 
of  the  mind ;  and  in  this  task,  it  can  aid  us  but  little  to 
call  it,  instead  of  the  mind,  the  subject,  or  the  me. 

SECT.  8. — Matter  and  Form. 

THERE  are  some  other  ways  of  expressing,  or  rather 
of  illustrating,  the  fundamental  antithesis,  which  I  may 
briefly  notice.  The  antithesis  has  been  at  different  times 
presented  by  means  of  various  images.  One  of  the  most 
ancient  of  these,  and  one  which  is  still  very  instructive, 
is  that  which  speaks  of  Sensations  as  the  Matter,  and 
Ideas  as  the  Form,  of  our  knowledge :  just  as  ivory  is 
VOL.  i.  w.  P.  D 

34  OF    IDKAS    IX    (JKXKUAL. 

the  matter,  and  a  cube  the  form,  of  a  die.  This  com 
parison  has  the  advantage  of  showing-  that  two  elements 
of  an  antithesis  which  cannot  be  separated  in  fact,  may 
yet  be  advantageously  separated  in  our  reasonings.  For 
Matter  and  Form  cannot  by  any  means  be  detached 
from  each  other.  All  matter  must  have  some  form ;  all 
form  must  be  the  form  of  some  material  thing.  If  the 
ivory  be  not  a  cube,  it  must  have  a  spherical  or  some 
other  farm.  And  the  cube,  in  order  to  be  a  cube,  must 
be  of  some  material ; — if  not  of  ivory,  of  wood,  or  stone, 
for  instance.  A  figure  without  matter  is  merely  a  geo 
metrical  conception ; — a  modification  of  the  idea  of 
space.  Matter  without  figure  is  a  mere  abstract  term ; 
—a  supposed  union  of  certain  sensible  qualities  which, 
so  insulated  from  others,  cannot  exist.  Yet  the  distinc 
tion  of  Matter  and  Form  is  real ;  and,  as  a  subject  of 
contemplation,  clear  and  plain.  Nor  is  the  distinction  by 
any  means  useless.  The  speculations  which  treat  of  the 
two  subjects,  Matter  and  Figure,  are  very  different. 
Matter  is  the  subject  of  the  sciences  of  Mechanics  and 
Chemistry  ;  Figure,  of  Geometry.  These  two  classes  of 
Sciences  have  quite  different  sets  of  principles.  If  we 
refuse  to  consider  the  Matter  and  the  Form  of  bodies 
separately,  because  we  cannot  exhibit  Matter  and  Form 
separately,  we  shut  the  door  to  all  philosophy  on  such 
subjects.  In  like  manner,  though  Sensations  and  Ideas 
are  necessarily  united  in  all  our  knowledge,  they  can  be 
considered  as  distinct;  and  this  distinction  is  the  basis  of 
all  philosophy  concerning  knowledge. 

This  illustration  of  the  relation  of  Ideas  and  Sensa 
tions  may  enable  us  to  estimate  a  doctrine  which  has  been 
put  forwards  at  various  times.  In  a  certain  school  of  spe 
culators  there  has  existed  a  disposition  to  derive  all  our 
Ideas  from  our  Sensations,  the  term  Idea  being,  in  this 
school,  used  in  its  wider  sense,  so  as  to  include  all  modifi- 


cations  and  limitations  of  our  Fundamental  Ideas.  The 
doctrines  of  this  school  have  been  summarily  expressed 
by  saying  that  "  Every  Idea  is  a  transformed  Sensation." 
Now,  even  supposing  this  assertion  to  be  exactly  true, 
we  easily  see,  from  what  has  been  said,  how  little  we 
are  likely  to  answer  the  ends  of  philosophy  by  putting 
forward  such  a  maxim  as  one  of  primary  importance. 
For  we  might  say,  in  like  manner,  that  every  statue  is 
but  a  transformed  block  of  marble,  or  every  edifice  but 
a  collection  of  transformed  stones.  But  what  would 
these  assertions  avail  us,  if  our  object  were  to  trace  the 
rules  of  art  by  which  beautiful  statues  were  formed,  or 
great  works  of  architecture  erected  ?  The  question 
naturally  occurs,  What  is  the  nature,  the  principle,  the 
law  of  this  Transformation  ?  In  what  faculty  resides  the 
transforming  power?  What  train  of  ideas  of  beauty, 
and  symmetry,  and  stability,  in  the  mind  of  the  statuary 
or  the  architect,  has  produced  those  great  works  which 
mankind  look  upon  as  among  their  most  valuable  pos 
sessions  ; — the  Apollo  of  the  Belvidere,  the  Parthenon, 
the  Cathedral  of  Cologne  ?  When  this  is  what  we  want 
to  know,  how  are  we  helped  by  learning  that  the  Apollo 
is  of  Parian  marble,  or  the  Cathedral  of  basaltic  stone  ? 
We  must  know  much  more  than  this,  in  order  to  acquire 
any  insight  into  the  principles  of  statuary  or  of  archi 
tecture.  In  like  manner,  in  order  that  we  may  make 
any  progress  in  the  philosophy  of  knowledge,  which  is 
our  purpose,  we  must  endeavour  to  learn  something 
further  respecting  ideas  than  that  they  are  transformed 
sensations,  even  if  they  were  this. 

But,  in  reality,  the  assertion  that  our  ideas  are  trans 
formed  sensations,  is  erroneous  as  well  as  frivolous.  For 
it  conveys,  and  is  intended  to  convey,  the  opinion  that 
our  sensations  have  one  form  which  properly  belongs  to 
them ;  and  that,  in  order  to  become  ideas,  they  are  con- 

D  2 

36  OF    1DKAS    IX    (JKNKKAL. 

verted  into  some  other  form.  But  the  truth  is,  that  our 
sensations,  of  themselves,  without  some  act  of  the  mind, 
such  as  involves  what  we  have  termed  an  Idea,  have  no 
form.  We  cannot  see  one  object  without  the  idea  of 
space ;  we  cannot  see  two  without  the  idea  of  resem 
blance  or  difference;  and  space  and  difference  are  not 
sensations.  Thus,  if  we  are  to  employ  the  metaphor  of 
Matter  and  Form,  which  is  implied  in  the  expression  to 
which  I  have  referred,  our  sensations,  from  their  first 
reception,  have  their  Form  not  changed,  but  given  by 
our  Ideas.  Without  the  relations  of  thought  which  we 
here  term  Ideas,  the  sensations  are  matter  without  form. 
Matter  without  form  cannot  exist :  and  in  like  manner 
sensations  cannot  become  perceptions  of  objects,  without 
some  formative  power  of  the  mind.  By  the  very  act  of 
being  received  as  perceptions,  they  have  a  formative 
power  exercised  upon  them,  the  operation  of  which 
might  be  expressed,  by  speaking  of  them,  not  as  trans 
formed,  but  simply  as  formed; — as  invested  with  form, 
instead  of  being  the  mere  formless  material  of  percep 
tion.  The  word  inform,  according  to  its  Latin  etymo 
logy,  at  first  implied  this  process  by  which  matter  is 
invested  with  form.  Thus  Virgil'-  speaks  of  the  thunder 
bolt  as  informed  by  the  hands  of  Brontes,  and  Steropes, 
and  Pyracmon.  And  Dryden  introduces  the  word  in 
another  place : — 

Let  others  better  mould  the  running  mass 
Of  metals,  or  inform  the  breathing  brass. 

Even  in  this  use  of  the  word,  the  form  is  something 
superior  to  the  brute  manner,  and  gives  it  a  new  signi 
ficance  and  purpose.  And  hence  the  term  is  again  used 

*  Ferrum  exercebant  vasto  Cyclopes  in  Antro 

Brontesque  Steropesque  et  nudus  membra  Pyracmon  ; 
His  informatum  manibus,  jam  parte  polita 
Fulmen  erat. — JEn.  viii.  424. 


to  denote  the  effect  produced  by  an  intelligent  principle 
of  a  still  higher  kind:— 

Hu  informed 

This  ill-shaped  body  with  a  daring  soul. 

And  finally  even  the  soul  itself,  in  its  original  condition, 
is  looked  upon  as  matter,  when  viewed  with  reference 
to  education  and  knowledge,  by  which  it  is  afterwards 
moulded ;  and  hence  these  are,  in  our  language,  termed 
information.  If  we  confine  ourselves  to  the  first  of 
these  three  uses  of  the  term,  we  may  correct  the  erro 
neous  opinion  of  which  we  have  just  been  speaking, 
and  retain  the  metaphor  by  which  it  is  expressed,  by 
saying,  that  ideas  are  not  transformed,  but  informed 

SECT.  9. — Man  the  Interpreter  of  Nature. 

THKRE  is  another  image  by  which  writers  have  repre 
sented  the  acts  of  thought  through  which  knowledge  is 
obtained  from  the  observation  of  the  external  world. 
Nature  is  the  Book,  and  Man  is  the  Interpreter.  The 
facts  of  the  external  world  are  marks,  in  which  man 
discovers  a  meaning,  and  so  reads  them.  Man  is  the 
Interpreter  of  Nature,  and  Science  is  the  right  Interpre 
tation.  And  this  image  also  is,  in  many  respects,  instruc 
tive.  It  exhibits  to  us  the  necessity  of  both  elements  ;— 
the  marks  which  man  has  to  look  at,  and  the  knowledge 
of  the  alphabet  and  language  which  he  must  possess  and 
apply  before  he  can  find  any  meaning  in  what  he  sees. 
Moreover  this  image  presents  to  us,  as  the  ideal  element, 
an  activity  of  the  mind  of  that  very  kind  which  we  wish 
to  point  out.  Indeed  the  illustration  is  rather  an 
example  than  a  comparison  of  the  composition  of  our 
knowledge.  The  letters  and  symbols  which  are  pre 
sented  to  the  Interpreter  are  really  objects  of  sensation  : 
the  notion  of  letters  as  signs  of  words,  the  notion  of 

38  OF    IDEAS   IN    GENERAL. 

connexions  among  words  by  which  they  have  meaning, 
really  are  among  our  Ideas ; — Signs  and  Meaning  are 
Ideas,  supplied  by  the  mind,  and  added  to  all  that  sensa 
tion  can  disclose  in  any  collection  of  visible  marks.  The 
Sciences  are  not  figuratively,  but  really,  Interpretations 
of  Nature.  But  this  image,  whether  taken  as  example  or 
comparison,  may  serve  to  show  both  the  opposite  charac 
ter  of  the  two  elements  of  knowledge,  and  their  neces 
sary  combination,  in  order  that  there  may  be  knowledge. 
This  illustration  may  also  serve  to  explain  another 
point  in  the  conditions  of  human  knowledge  which  we 
shall  have  to  notice  : — namely,  the  very  different  degrees 
in  which,  in  different  cases,  we  are  conscious  of  the 
mental  act  by  which  our  sensations  are  converted  into 
knowledge.  For  the  same  difference  occurs  in  reading 
an  inscription.  If  the  inscription  were  entire  and  plain, 
in  a  language  with  which  we  were  familiar,  we  should 
be  unconscious  of  any  mental  act  in  reading  it.  We 
should  seem  to  collect  its  meaning  by  the  sight  alone. 
But  if  we  had  to  decipher  an  ancient  inscription,  of 
which  only  imperfect  marks  remained,  with  a  few  entire 
letters  among  them,  we  should  probably  make  several 
suppositions  as  to  the  mode  of  reading  it,  before  we 
found  any  mode  which  was  quite  successful ;  and  thus, 
our  guesses,  being  separate  from  the  observed  facts,  and 
at  first  not  fully  in  agreement  with  them,  we  should  be 
clearly  aware  that  the  conjectured  meaning,  on  the  one 
hand,  and  the  observed  marks  on  the  other,  were  dis 
tinct  things,  though  these  two  things  would  become 
united  as  elements  of  one  act  of  knowledge  when  we 
had  hit  upon  the  right  conjecture. 

SECT.  10. — The  Fundamental  Antithesis  inseparable. 

THE    illustration  just    referred    to,    as    well    as    other 
ways  of  considering  the  subject,  may  help  us  to  get  over 


a  difficulty  which  at  first  sight  appears  perplexing.  We 
have  spoken  of  the  common  opposition  of  Theory  and 
Fact  as  important,  and  as  involving  what  we  have  called 
the  Fundamental  Antithesis  of  Philosophy.  But  after 
all,  it  may  be  asked.  Is  this  distinction  of  Theory  and 
Fact  really  tenable?  Is  it  not  often  difficult  to  say 
whether  a  special  part  of  our  knowledge  is  a  Fact  or 
a  Theory?  Is  it  a  Fact  or  a  Theory  that  the  stars 
revolve  round  the  pole?  Is  it  a  Fact  or  a  Theory  that 
the  earth  is  a  globe  revolving  on  its  axis?  Is  it  a  Fact 
or  a  Theory  that  the  earth  travels  in  an  ellipse  round 
the  sun?  Is  it  a  Fact  or  a  Theory  that  the  sun  attracts 
the  earth?  Is  it  a  Fact  or  a  Theory  that  the  loadstone 
attracts  the  needle?  In  all  these  cases,  probably  some 
persons  would  answer  one  way,  and  some  persons  the 
other.  There  are  many  persons  by  whom  the  doctrine 
of  the  globular  form  of  the  earth,  the  doctrine  of  the 
earth's  elliptical  orbit,  the  doctrine  of  the  sun's  attrac 
tion  on  the  earth,  would  be  called  theories,  even  if  they 
allowed  them  to  be  true  theories.  But  yet  if  each  of 
these  propositions  be  true,  is  it  not  &factf  And  even 
with  regard  to  the  simpler  facts,  as  the  motion  of  the 
stars  round  the  pole,  although  this  may  be  a  Fact  to  one 
who  has  watched  and  measured  the  motions  of  the  stars, 
one  who  has  not  done  this,  and  who  has  only  carelessly 
looked  at  these  stars  from  time  to  time,  may  naturally 
speak  of  the  circles  which  the  astronomer  makes  them 
describe  as  Theories.  It  would  seem,  then,  that  we 
cannot  in  such  cases  expect  general  assent,  if  we  say, 
This  is  a  Fact  and  not  a  Theory,  or,  This  is  a  Theory 
and  not  a  Fact.  And  the  same  is  true  in  a  vast  range 
of  cases.  It  would  seem,  therefore,  that  we  cannot  rest 
any  reasoning  upon  this  distinction  of  Theory  and  Fact: 
and  we  cannot  avoid  asking  whether  there  is  any  real 
distinction  in  this  antithesis,  and  if  so.  what  it  is. 

40  OF    IDEAS    IN    GENERAL. 

To  this  I  reply :  the  distinction  between  Theory 
(that  is,  true  Theory)  and  Fact,  is  this:  that  in  Theory 
the  Ideas  are  considered  as  distinct  from  the  Facts:  in 
Facts,  though  Ideas  may  be  involved,  they  are  not,  in 
our  apprehension,  separated  from  the  sensations.  In  a 
Fact,  the  Ideas  are  applied  so  readily  and  familiarly,  and 
incorporated  with  the  sensations  so  entirely,  that  we 
do  not  see  them,  we  see  through  them.  A  person  who 
carefully  notes  the  motion  of  a  star  all  night,  sees  the 
circle  which  it  describes,  as  he  sees  the  star,  though 
the  circle  is,  in  fact,  a  result  of  his  own  Ideas.  A 
person  who  has  in  his  mind  the  measures  of  different 
lines  and  countries  on  the  earth's  surface,  and  who  can 
put  them  together  into  one  conception,  finds  that  they 
can  make  no  figure  but  a  globular  one:  to  him,  the 
earth's  globular  form  is  a  Fact,  as  much  as  the  square 
form  of  his  chamber.  A  person  to  whom  the  grounds 
of  believing  the  earth  to  travel  round  the  sun  are  as 
familiar  as  the  grounds  for  believing  the  movements 
of  the  mail-coaches  in  this  country,  looks  upon  the 
former  event  as  a  Fact,  just  as  he  looks  upon  the  latter 
events  as  Facts.  And  a  person  who,  knowing  the  Fact 
of  the  earth's  annual  motion,  refers  it  distinctly  to  its 
mechanical  cause,  conceives  the  sun's  attraction  as  a 
Fact,  just  as  he  conceives  as  a  Fact,  the  action  of  the 
wind  which  turns  the  sails  of  a  mill.  He  cannot  see 
the  force  in  either  case ;  he  supplies  it  out  of  his  own 
Ideas.  And  thus,  a  true  Theory  is  a  Fact ;  a  Fact  is 
a  familiar  Theory.  That  which  is  a  Fact  under  one 
aspect,  is  a  Theory  under  another.  The  most  recondite 
Theories  when  firmly  established  are  Facts:  the  sim 
plest  Facts  involve  something  of  the  nature  of  Theory. 
Theory  and  Fact  correspond,  in  a  certain  degree,  with 
Ideas  and  Sensations,  as  to  the  nature  of  their  opposi 
tion.  But  the  Facts  are  Facts,  so  far  as  the  Ideas  have 


been  combined  with  the  Sensations  and  absorbed  in 
them:  the  Theories  are  Theories,  so  far  as  the  Ideas 
are  kept  distinct  from  the  Sensations,  and  so  far  as  it  is 
considered  still  a  question  whether  those  can  be  made 
to  agree  with  these. 

We  may,  as  I  have  said,  illustrate  this  matter  by 
considering  man  as  interpreting  the  phenomena  which 
he  sees.  He  often  interprets  without  being  aware  that 
he  does  so.  Thus  when  we  see  the  needle  move  towards 
the  magnet,  we  assert  that  the  magnet  exercises  an 
attractive  force  on  the  needle.  But  it  is  only  by  an 
interpretative  act  of  our  own  minds  that  we  ascribe 
this  motion  to  attraction.  That,  in  this  case,  a  force  is 
exerted — something  of  the  nature  of  the  pull  which  we 
could  apply  by  our  own  volition — is  our  interpretation 
of  the  phenomena;  although  we  may  be  conscious  of  the 
act  of  interpretation,  and  may  then  regard  the  attrac 
tion  as  a  Fact. 

Nor  is  it  in  such  cases  only  that  we  interpret  phe 
nomena  in  our  own  way,  without  being  conscious  of 
what  we  do.  We  see  a  tree  at  a  distance,  and  judge  it 
to  be  a  chestnut  or  a  lime ;  yet  this  is  only  an  inference 
from  the  colour  or  form  of  the  mass  according  to  pre 
conceived  classifications  of  our  own.  Our  lives  are  full 
of  such  unconscious  interpretations.  The  farmer  recog 
nizes  a  good  or  a  bad  soil ;  the  artist  a  picture  of  a 
favourite  master ;  the  geologist  a  rock  of  a  known  local 
ity,  as  we  recognize  the  faces  and  voices  of  our  friends ; 
that  is,  by  judgments  formed  on  what  we  see  and  hear : 
but  judgments  in  which  we  do  not  analyze  the  steps,  or 
distinguish  the  inference  from  the  appearance.  And  in 
these  mixtures  of  observation  and  inference,  we  speak  of 
the  judgment  thus  formed,  as  a  Fact  directly  observed. 

Even  in  the  case  in  which  our  perceptions  appear  to 
be  most  direct,  and  least  to  involve  any  interpretations 

42  OF    IDEAS    IN    GENERAL. 

of  our  own, — in  the  simple  process  of  seeing, — who  does 
not  know  how  much  we,  by  an  act  of  the  mind,  add  to 
that  which  our  senses  receive  ?  Does  any  one  fancy  that 
lie  sees  a  solid  cube  ?  It  is  easy  to  show  that  the  solid 
ity  of  the  figure,  the  relative  position  of  its  faces  and 
edges  to  each  other,  are  inferences  of  the  spectator ;  no 
more  conveyed  to  his  conviction  by  the  eye  alone,  than 
they  would  be  if  he  were  looking  at  a  painted  represen 
tation  of  a  cube.  The  scene  of  nature  is  a  picture  with 
out  depth  of  substance,  no  less  than  the  scene  of  art ; 
and  in  the  one  case  as  in  the  other,  it  is  the  mind  which, 
by  an  act  of  its  own,  discovers  that  colour  and  shape 
denote  distance  and  solidity.  Most  men  are  unconscious 
of  this  perpetual  habit  of  reading  the  language  of  the 
external  world,  and  translating  as  they  read.  The 
draughtsman,  indeed,  is  compelled,  for  his  purposes,  to 
return  back  in  thought  from  the  solid  bodies  which  he 
has  inferred,  to  the  shapes  of  surface  which  he  really 
sees.  He  knows  that  there  is  a  mask  of  theory  over  the 
whole  face  of  nature,  if  it  be  tlieonj  to  infer  more  than 
we  see.  But  other  men,  unaware  of  this  masquerade, 
hold  it  to  be  a  fact  that  they  see  cubes  and  spheres,  spa 
cious  apartments  and  winding  avenues.  And  these  things 
are  facts  to  them,  because  they  are  unconscious  of  the 
mental  operation  by  which  they  have  penetrated  nature's 

And  thus,  we  still  have  an  intelligible  distinction  of 
Fact  and  Theory,  if  we  consider  Theory  as  a  conscious,  and 
Fact  as  an  unconscious  inference,  from  the  phenomena 
which  are  presented  to  our  senses. 

But  still,  Theory  and  Fact,  Inference  and  Perception, 
Reasoning  and  Observation,  are  antitheses  in  none  of 
which  can  we  separate  the  two  members  by  any  fixed 
and  definite  line. 

Even  the  simplest  terms  by  which  the  antithesis  is 


expressed  cannot  be  separated.  Ideas  and  Sensations, 
Thoughts  and  Things,  Subject  and  Object,  cannot  in  any 
case  be  applied  absolutely  and  exclusively.  Our  Sen 
sations  require  Ideas  to  bind  them  together,  namely, 
Ideas  of  space,  time,  number,  and  the  like.  If  not  so 
bound  together,  Sensations  do  not  give  us  any  appre 
hension  of  Things  or  Objects.  All  Things,  all  Objects, 
must  exist  in  space  and  in  time — must  be  one  or  many. 
Now  space,  time,  number,  are  not  Sensations  or  Things. 
They  are  something  different  from,  and  opposed  to  Sen 
sations  and  Things.  We  have  termed  them  Ideas.  It 
may  be  said  they  are  Relations  of  Things,  or  of  Sensa 
tions.  But  granting  this  form  of  expression,  still  a 
Relation  is  not  a  Thing  or  a  Sensation ;  and  therefore 
we  must  still  have  another  and  opposite  element,  along 
with  our  Sensations.  And  yet,  though  we  have  thus 
these  two  elements  in  every  act  of  perception,  we  cannot 
designate  any  portion  of  the  act  as  absolutely  and  exclu 
sively  belonging  to  one  of  the  elements.  Perception 
involves  Sensation,  along  with  Ideas  of  time,  space,  and 
the  like ;  or,  if  any  one  prefers  the  expression,  we  may 
say,  Perception  involves  Sensations  along  with  the  ap 
prehension  of  Relations.  Perception  is  Sensation,  along 
with  such  Ideas  as  make  Sensation  into  an  apprehension 
of  Things  or  Objects. 

And  as  Perception  of  Objects  implies  Ideas, — as  Ob 
servation  implies  Reasoning; — so,  on  the  other  hand. 
Ideas  cannot  exist  where  Sensation  has  not  been ;  Rea 
soning  cannot  go  on  when  there  has  not  been  previous 
Observation.  This  is  evident  from  the  necessary  order 
of  developement  of  the  human  faculties.  Sensation 
necessarily  exists  from  the  first  moments  of  our  exist 
ence,  and  is  constantly  at  work.  Observation  begins 
before  we  can  suppose  the  existence  of  any  Reasoning 
which  is  not  involved  in  Observation.  Hence,  at  what- 

44  OF    IDEAS    IN    GENERAL. 

ever  period  we  consider  our  Ideas,  we  must  consider 
them  as  having  been  already  engaged  in  connecting  our 
Sensations,  and  as  having  been  modified  by  this  employ 
ment.  By  being  so  employed,  our  Ideas  are  unfolded 
and  defined  ;  and  such  developement  and  definition  can 
not  be  separated  from  the  Ideas  themselves.  We  cannot 
conceive  space,  without  boundaries  or  forms ;  now  Forms 
involve  Sensations.  We  cannot  conceive  time,  without 
events  which  mark  the  course  of  time  ;  but  events  involve 
Sensations.  We  cannot  conceive  number,  without  con 
ceiving  things  which  are  numbered ;  and  Things  imply 
sensations.  And  the  forms,  things,  events,  which  are 
thus  implied  in  our  Ideas,  having  been  the  objects  of 
Sensation  constantly  in  every  part  of  our  life,  have 
modified,  unfolded,  and  fixed  our  Ideas,  to  an  extent 
which  we  cannot  estimate,  but  which  we  must  suppose 
to  be  essential  to  the  processes  which  at  present  go  on 
in  our  minds.  We  cannot  say  that  Objects  create  Ideas ; 
for  to  perceive  Objects  we  must  already  have  Ideas. 
But  we  may  say,  that  Objects  and  the  constant  Perception 
of  Objects  have  so  far  modified  our  Ideas,  that  we  cannot, 
oven  in  thought,  separate  our  Ideas  from  the  perception 
of  Objects. 

We  cannot  say  of  any  Ideas,  as  of  the  Idea  of  space, 
or  time,  or  number,  that  they  are  absolutely  and  exclu 
sively  Ideas.  We  cannot  conceive  what  space,  or  time, 
or  number,  would  be  in  our  minds,  if  we  had  never  per 
ceived  any  Thing  or  Things  in  space  or  time.  We  can 
not  conceive  ourselves  in  such  a  condition  as  never  to  have 
perceived  any  Thing  or  Things  in  space  or  time.  But,  on 
the  other  hand,  just  as  little  can  we  conceive  ourselves 
becoming  acquainted  with  space  and  time  or  numbers 
as  objects  of  Sensation.  We  cannot  reason  without 
having  the  operations  of  our  minds  affected  by  previous 
Sensations ;  but  we  cannot  conceive  Reasoning  to  be 


merely  a  series  of  Sensations.  In  order  to  be  used  in 
Reasoning,  Sensation  must  become  Observation ;  and,  as 
we  have  seen,  Observation  already  involves  Reasoning. 
In  order  to  be  connected  by  our  Ideas,  Sensations  must 
be  Things  or  Objects,  and  Things  or  Objects  already  in 
clude  Ideas.  And  thus,  none  of  the  terms  by  which  the 
fundamental  antithesis  is  expressed  can  be  absolutely 
and  exclusively  applied. 

I  will  make  a  remark  suggested  by  the  views  which 
have  thus  been  presented.     Since,  as  we  have  just  seen, 
none  of  the  terms  which  express  the  fundamental  anti 
thesis  can  be  applied   absolutely  and    exclusively,   the 
absolute  application  of  the  antithesis  in  any  particular 
case  can  never  be  a  conclusive  or  immoveable  principle. 
This  remark  is  the  more  necessary  to  be  borne  in  mind,  as 
the  terms  of  this  antithesis  are  often  used  in  a  vehement 
and  peremptory  manner.     Thus  we  are  often  told  that 
such  a  thing  is  a  Fact;  A  FACT  and  not  a  Theory,  with  all 
the  emphasis  which,  in  speaking  or  writing,  tone  or  italics 
or  capitals  can  give.     We  see  from  what  has  been  said, 
that  when  this  is   urged,    before    we  can  estimate  the 
truth,  or  the  value  of  the  assertion,    we   must  ask   to 
whom  is  it  a  Fact?  what  habits  of  thought,  what  pre 
vious  information,  what  Ideas  does  it  imply,  to  conceive 
the  Fact  as  a  Fact  ?     Does  not  the  apprehension  of  the 
Fact  imply  assumptions  which  may  with  equal  justice 
be  called  Theory,  and  which  are  perhaps  false  Theory? 
in  which  case,  the  Fact  is  no  Fact.     Did  not  the  an 
cients  assert   it  as  a  Fact,  that  the   earth  stood  still, 
and  the  stars  moved  ?  and  can  any  Fact  have  stronger 
apparent  evidence  to  justify  persons  in  asserting  it  em 
phatically  than  this  had  ? 

These  remarks  are  by  no  means  urged  in  order  to 
shew  that  no  Fact  can  be  certainly  known  to  be  true ; 
but  onlv,  to  shew  that  no  Fact  can  be  certainlv  shown 

4(i  OF    IDEAS    IN    GENERAL. 

to  be  a  Fact,  merely  by  calling  it  a  Fact,  however 
emphatically.  There  is  by  no  means  any  ground  of 
general  skepticism  with  regard  to  truth,  involved  in 
the  doctrine  of  the  necessary  combination  of  two  ele 
ments  in  all  our  knowledge.  On  the  contrary,  Ideas 
are  requisite  to  the  essence,  and  Things  to  the  reality 
of  our  knowledge  in  every  case.  The  proportions  of 
Geometry  and  Arithmetic  are  examples  of  knowledge 
respecting  our  Ideas  of  space  and  number,  with  regard 
to  which  there  is  no  room  for  doubt.  The  doctrines  of 
Astronomy  are  examples  of  truths  not  less  certain 
respecting  the  Facts  of  the  external  world. 

SECT.  11. — Successive  Generalization. 

IN  the  preceding  pages  we  have  been  led  to  the  doctrine, 
that  though,  in  the  Antithesis  of  Theory  and  Fact,  there 
is  involved  an  essential  opposition ;  namely  the  opposition 
of  the  thoughts  within  us  and  the  phenomena  without 
us ;  yet  that  we  cannot  distinguish  and  define  the  mem 
bers  of  this  antithesis  separately.  Theories  become 
Facts,  by  becoming  certain  and  familiar :  and  thus,  as 
our  knowledge  becomes  more  sure  and  more  extensive, 
we  are  constantly  transferring  to  the  class  of  facts, 
opinions  which  were  at  first  regarded  as  theories. 

Now  we  have  further  to  remark,  that  in  the  progress 
of  human  knowledge  respecting  any  branch  of  specula 
tion,  there  may  be  several  such  steps  in  succession,  each 
depending  upon  and  including  the  preceding.  The 
theoretical  views  which  one  generation  of  discoverers 
establishes,  become  the  facts  from  which  the  next  gene 
ration  advances  to  new  theories.  As  men  rise  from  the 
particular  to  the  general,  so,  in  the  same  manner,  they 
rise  from  what  is  general  to  what  is  more  general.  Each 
induction  supplies  the  materials  of  fresh  inductions ; 
each  generalization,  with  all  that  it  embraces  in  its  circle. 


may  be  found  to  be  but  one  of  many  circles,  compre 
hended  within  the  circuit  of  some  wider  generalization. 

This  remark  has  already  been  made,  and  illustrated, 
in  the  History  of  the  Inductive  Sciences"' ;  and,  in  truth, 
the  whole  of  the  history  of  science  is  full  of  suggestions 
and  exemplifications  of  this  course  of  things.  It  may  be 
convenient,  however,  to  select  a  few  instances  which  may 
further  explain  and  confirm  this  view  of  the  progress  of 
scientific  knowledge. 

The  most  conspicuous  instance  of  this  succession  is 
to  be  found  in  that  science  which  has  been  progressive 
from  the  beginning  of  the  world  to  our  own  times,  and 
which  exhibits  by  far  the  richest  collection  of  successive 
discoveries :  I  mean  Astronomy.  It  is  easy  to  see  that 
each  of  these  successive  discoveries  depended  on  those 
antecedently  made,  and  that  in  each,  the  truths  which 
were  the  highest  point  of  the  knowledge  of  one  age 
were  the  fundamental  basis  of  the  efforts  of  the  age 
which  came  next.  Thus  we  find,  in  the  days  of  Greek 
discovery,  Hipparchus  and  Ptolemy  combining  and  ex 
plaining  the  particular  facts  of  the  motion  of  the  sun, 
moon,  and  planets,  by  means  of  the  theory  of  epicycles 
and  eccentrics  ; — a  highly  important  step,  which  gave 
an  intelligible  connexion  and  rule  to  the  motions  of  each 
of  these  luminaries.  When  these  cycles  and  epicycles, 
thus  truly  representing  the  apparent  motions  of  the 
heavenly  bodies,  had  accumulated  to  an  inconvenient 
amount,  by  the  discovery  of  many  inequalities  in  the 
observed  motions,  Copernicus  showed  that  their  effects 
might  all  be  more  simply  included,  by  making  the  sun 
the  center  of  motion  of  the  planets,  instead  of  the  earth. 
But  in  this  new  view,  he  still  retained  the  epicycles  and 
eccentrics  which  governed  the  motion  of  each  body. 
Tycho  Brahe's  observations,  and  Kepler's  calculations, 

f   Uixf.  Inductive  Sciences,  B.  vn    c.  ii.  Sort.  "». 

48  OF    IDEAS    JN    GENERAL. 

showed  that,  besides  the  vast  number  of  facts  which  the 
epicyclical  theory  could  account  for,  there  were  some 
which  it  would  not  exactly  include,  and  Kepler  was  led 
to  the  persuasion  that  the  planets  move  in  ellipses. 
But  this  view  of  motion  was  at  first  conceived  by  Kepler 
as  a  modification  of  the  conception  of  epicycles.  On  one 
occasion  he  blames  himself  for  not  sooner  seeing  that 
such  a  modification  was  possible.  "  What  an  absurdity 
on  my  part !"  he  cries"'- ;  "  as  if  libration  in  the  diameter 
of  the  epicycle  might  not  come  to  the  same  thing  as 
motion  in  the  ellipse."  But  again;  Kepler's  laws  of  the 
elliptical  motion  of  the  planets  were  established ;  and 
these  laws  immediately  became  the  facts  on  which  the 
mathematicians  had  to  found  their  mechanical  theories. 
From  these  facts,  Newton,  as  we  have  related,  proved 
that  the  central  force  of  the  sun  retains  the  planets  in 
their  orbits,  according  to  the  law  of  the  inverse  square 
of  the  distance.  The  same  law  was  shown  to  prevail  in 
the  gravitation  of  the  earth.  It  was  shown,  too,  by  in 
duction  from  the  motions  of  Jupiter  and  Saturn,  that 
the  planets  attract  each  other ;  by  calculations  from  the 
figure  of  the  earth,  that  the  parts  of  the  earth  attract 
each  other ;  and,  by  considering  the  course  of  the  tides, 
that  the  sun  and  moon  attract  the  waters  of  the  ocean. 
And  all  these  curious  discoveries  being  established  as 
facts,  the  subject  was  ready  for  another  step  of  gene 
ralization.  By  an  unparalleled  rapidity  in  the  progress 
of  discovery  in  this  case,  not  only  were  all  the  inductions 
which  we  have  first  mentioned  made  by  one  individual, 
but  the  new  advance,  the  higher  flight,  the  closing  vic 
tory,  fell  to  the  lot  of  the  same  extraordinary  person. 

The  attraction  of  the  sun  upon  the  planets,  of  the 
moon  upon  the  earth,  of  the  planets  on  each  other,  of  the 
parts  of  the  earth  on  themselves,  of  the  sun  and  moon 

*   Hist.  Inductive  Sciences,  B.  v.  c.  iv.  Sect.  3. 


upon  the  ocean; — all  these  truths,  each  of  itself  a  great 
discovery,  were  included  by  Newton  in  the  higher  gene 
ralization,  of  the  universal  gravitation  of  matter,  by 
which  each  particle  is  drawn  to  each  other  according  to 
the  law  of  the  inverse  square :  and  thus  this  long  ad 
vance  from  discovery  to  discovery,  from  truths  to  truths, 
each  justly  admired  when  new,  and  then  rightly  used  as 
old,  was  closed  in  a  worthy  and  consistent  manner,  by 
a  truth  which  is  the  most  worthy  admiration,  because  it 
includes  all  the  researches  of  preceding  ages  of  Astro 

We  may  take  another  example  of  a  succession  of  this 
kind  from  the  history  of  a  science,  which,  though  it  has 
made  wonderful  advances,  has  not  yet  reached  its  goal, 
as  physical  astronomy  appears  to  have  done,  but  seems  to 
have  before  it  a  long  prospect  of  future  progress.  I  now 
refer  to  Chemistry,  in  which  I  shall  try  to  point  out  how 
the  preceding  discoveries  afforded  the  materials  of  the 
succeeding;  although  this  subordination  and  connexion 
is,  in  this  case,  less  familiar  to  men's  minds  than  in  Astro 
nomy,  and  is,  perhaps,  more  difficult  to  present  in  a  clear 
and  definite  shape.  Sylvius  saw.  in  the  facts  which 
occur,  when  an  acid  and  an  alkali  are  brought  together, 
the  evidence  that  they  neutralize  each  other.  But  cases 
of  neutralization,  and  acidification,  and  many  other  ef 
fects  of  mixture  of  the  ingredients  of  bodies,  being  thus 
viewed  as  facts,  had  an  aspect  of  unity  and  law  given 
them  by  Geoffroy  and  Bergman""",  who  introduced  the  con 
ception  of  the  Chemical  Affinity  or  Elective  Attraction, 
by  which  certain  elements  select  other  elements,  as  if  by 
preference.  That  combustion,  whether  a  chemical  union 
or  a  chemical  separation  of  ingredients,  is  of  the  same 
nature  with  acidification,  was  the  doctrine  of  Beccher 

*   Hixt.  Inductive  Sciences,  B.  xiv.  c.  iii. 
VOL.  I.     W.  P.  E 

•30  OF    IDEAS    IN    GENERAL. 

and  Stahl,  and  was  soon  established  as  a  truth  which 
must  form  a  part  of  every  succeeding  physical  theory. 
That  the  rules  of  affinity  and  chemical  composition  may 
include  gaseous  elements,  was  established  by  Black  and 
Cavendish.  And  all  these  truths,  thus  brought  to  light 
by  chemical  discoverers, — affinity,  the  identity  of  acidifi 
cation  and  combustion,  the  importance  of  gaseous  ele 
ments, — along  with  all  the  facts  respecting  the  weight 
of  ingredients  and  compounds  which  the  balance  dis 
closed, — were  taken  up,  connected,  and  included  as 
particulars  in  the  oxygen  theory  of  Lavoisier.  Again, 
the  results  of  this  theory,  and  the  quantity  of  the  several 
ingredients  which  entered  into  each  compound — (such 
results,  for  the  most  part,  being  now  no  longer  mere 
theoretical  speculations,  but  recognized  facts) — were  the 
particulars  from  which  Dalton  derived  that  wide  law  of 
chemical  combination  which  we  term  the  Atomic  Theory. 
And  this  law,  soon  generally  accepted  among  chemists, 
is  already  in  its  turn  become  one  of  the  facts  included 
in  Faraday's  Theory  of  the  identity  of  Chemical  Affinity 
and  Electric  Attraction. 

It  is  unnecessary  to  give  further  exemplifications  of 
this  constant  ascent  from  one  step  to  a  higher ; — this 
perpetual  conversion  of  true  theories  into  the  materials 
of  other  and  wider  theories.  It  will  hereafter  be  our 
business  to  exhibit,  in  a  more  full  and  formal  manner, 
the  mode  in  which  this  principle  determines  the  whole 
scheme  and  structure  of  all  the  most  exact  sciences. 
And  thus,  beginning  with  the  facts  of  sense,  we  gradually 
climb  to  the  highest  forms  of  human  knowledge,  and 
obtain  from  experience  and  observation  a  vast  collection 
of  the  most  wide  and  elevated  truths. 

There  are,  however,  truths  of  a  very  different  kind,  to 
which  we  must  turn  our  attention,  in  order  to  pursue  our 


researches  respecting  the  nature  and  grounds  of  our 
knowledge.  But  before  we  do  this,  we  must  notice  one 
more  feature  in  that  progress  of  science  which  we  have 
already  in  part  described. 


1 .  IT  has  already  been  stated  that  we  gather  knowledge 
from  the  external  world,  when  we  are  able  to  apply,  to 
the  facts  which  we  observe,  some  ideal  conception,  which 
gives  unity  and  connexion  to  multiplied  and  separate 
perceptions.  We  have  also  shown  that  our  conceptions, 
thus  verified  by  facts,  may  themselves  be  united  and  con 
nected  by  a  new  bond  of  the  same  nature ;  and  that  man 
may  thus  have  to  pursue  his  way  from  truth  to  truth 
through  a  long  progression  of  discoveries,  each  resting 
on  the  preceding,  and  rising  above  it. 

Each  of  these  steps,  in  succession,  is  recorded,  fixed, 
and  made  available,  by  some  peculiar  form  of  words ; 
and  such  words,  thus  rendered  precise  in  their  meaning, 
and  appropriated  to  the  service  of  science,  we  may  call 
Technical  Terms.  It  is  in  a  great  measure  by  inventing 
such  Terms  that  men  not  only  best  express  the  discoveries 
they  have  made,  but  also  enable  their  followers  to  become 
so  familiar  with  these  discoveries,  and  to  possess  them 
so  thoroughly,  that  they  can  readily  use  them  in  ad 
vancing  to  ulterior  generalizations. 

Most  of  our  ideal  conceptions  are  described  by  exact 
and  constant  words  or  phrases,  such  as  those  of  which  we 
here  speak.  We  have  already  had  occasion  to  employ 
many  of  these.  Thus  we  have  had  instances  of  technical 
Terms  expressing  geometrical  conceptions,  as  Ellipsis, 



Radius  Vector,  Axis,  Plane,  the  Proportion  of  the  In 
verse  Square,  and  the  like.  Other  Terms  have  described 
mechanical  conceptions,  as  Accelerating  Force  and 
Attraction.  Again,  chemistry  exhibits  (as  do  all  sciences) 
a  series  of  Terms  which  mark  the  steps  of  our  progress. 
The  views  of  the  first  real  founders  of  the  science  are 
recorded  by  the  Terms  which  are  still  in  use,  Neutral 
Salts,  Affinity,  and  the  like.  The  establishment  of  Dai- 
ton's  theory  has  produced  the  use  of  the  word  Atom  in 
a  peculiar  sense,  or  of  some  other  word,  as  Proportion, 
in  a  sense  equally  technical.  And  Mr.  Faraday  has 
found  it  necessary,  in  order  to  expound  his  electro-chemi 
cal  theory,  to  introduce  such  terms  as  Anode  and  Cathode, 
Anion  and  CatMon. 

2.  I  need  not  adduce  any  further  examples,  for  my 
object  at  present  is  only  to  point  out  the  use  and  influence 
of  such  language :  its  rules  and  principles  I  shall  here 
after  try,  in  some  measure,  to  fix.  But  what  we  have 
here  to  remark  is,  the  extraordinary  degree  in  which  the 
progress  of  science  is  facilitated,  by  thus  investing  each 
new  discovery  with  a  compendious  and  steady  form  of 
expression.  These  terms  soon  become  part  of  the  cur 
rent  language  of  all  who  take  an  interest  in  speculation. 
However  strange  they  may  sound  at  first,  they  soon  grow 
familiar  in  our  ears,  and  are  used  without  any  effort,  or 
any  recollection  of  the  difficulty  they  once  involved.  They 
become  as  common  as  the  phrases  which  express  our 
most  frequent  feelings  and  interests,  while  yet  they  have 
incomparably  more  precision  than  belongs  to  any  terms 
which  express  feelings;  and  they  carry  with  them,  in 
their  import,  the  results  of  deep  and  laborious  trains  of 
research.  They  convey  the  mental  treasures  of  one 
period  to  the  generations  that  follow ;  and  laden  with 
this,  their  precious  freight,  they  sail  safely  across  gulfs 
of  time  in  which  empires  have  suffered  shipwreck,  and 


the  languages  of  common  life  have  sunk  into  oblivion. 
We  have  still  in  constant  circulation  among  us  the  Terms 
which  belong  to  the  geometry,  the  astronomy,  the 
zoology,  the  medicine  of  the  Greeks,  and  the  algebra 
and  chemistry  of  the  Arabians.  And  we  can  in  an  in 
stant,  by  means  of  a  few  words,  call  to  our  own  recollec 
tion,  or  convey  to  the  apprehension  of  another  person, 
phenomena  and  relations  of  phenomena  in  optics,  mine 
ralogy,  chemistry,  which  are  so  complex  and  abstruse, 
that  it  might  seem  to  require  the  utmost  subtlety  of  the 
human  mind  to  grasp  them,  even  if  that  were  made  the 
sole  object  of  its  efforts.  By  this  remarkable  effect  of 
Technical  Language,  we  have  the  results  of  all  the 
labours  of  past  times  not  only  always  accessible,  but  so 
prepared  that  we  may  (provided  we  are  careful  in  the 
use  of  our  instrument)  employ  what  is  really  useful  and 
efficacious  for  the  purpose  of  further  success,  without 
being  in  any  way  impeded  or  perplexed  by  the  length 
and  weight  of  the  chain  of  past  connexions  which  we 
drag  along  with  us. 

By  such  means, — by  the  use  of  the  Inductive  Process, 
and  by  the  aid  of  Technical  Terms, — man  has  been  con 
stantly  advancing  in  the  path  of  scientific  truth.  In  a 
succeeding  part  of  this  work  we  shall  endeavour  to  trace 
the  general  rules  of  this  advance,  and  to  lay  down  the 
maxims  by  which  it  may  be  most  successfully  guided 
and  forwarded.  But  in  order  that  we  may  do  this  to 
the  best  advantage,  we  must  pursue  still  further  the 
analysis  of  knowledge  into  its  elements ;  and  this  will  be 
our  employment  in  the  first  part  of  the  work. 



1 .  EVERY  advance  in  human  knowledge  consists,  as 
we  have  seen,  in  adapting  new  ideal  conceptions  to  ascer 
tained  facts,  and  thus  in  superinducing  the  Form  upon 
the  Matter,  the  active  upon  the  passive  processes  of  our 
minds.  Every  such  step  introduces  into  our  knowledge 
an  additional  portion  of  the  ideal  element,  and  of  those 
relations  which  flow  from  the  nature  of  Ideas.  It  is, 
therefore,  important  for  our  purpose  to  examine  more 
closely  this  element,  and  to  learn  what  the  relations  are 
which  may  thus  come  to  form  part  of  our  knowledge. 
An  inquiry  into  those  Ideas  which  form  the  foundations 
of  our  sciences ; — into  the  reality,  independence,  extent, 
and  principal  heads  of  the  knowledge  which  we  thus  ac 
quire  ; — is  a  task  on  which  we  must  now  enter,  and 
which  will  employ  us  for  several  of  the  succeeding  Books. 

In  this  inquiry  our  object  will  be  to  pass  in  review  all 
the  most  important  Fundamental  Ideas  which  our 
sciences  involve ;  and  to  prove  more  distinctly  in  refer 
ence  to  each,  what  we  have  already  asserted  with  regard 
to  all,  that  there  are  everywhere  involved  in  our  know 
ledge  acts  of  the  mind  as  well  as  impressions  of  sense ; 
and  that  our  knowledge  derives,  from  these  acts,  a  gene 
rality,  certainty,  and  evidence  which  the  senses  could  in 
no  degree  have  supplied.  But  before  I  proceed  to  do 
this  in  particular  cases,  I  will  give  some  account  of  the 
argument  in  its  general  form. 

We  have  already  considered  the  separation  of  our 
knowledge  into  its  two  elements, — Impressions  of  Sense 
and  Ideas, — as  evidently  indicated  by  this ;  that  all  know 
ledge  possesses  characters  which  neither  of  these  ele 
ments  alone  could  bestow.  Without  our  ideas,  our  sen 
sations  could  have  no  connexion ;  without  external 


impressions,  our  ideas  would  have  no  reality ;  and  thus 
both  ingredients  of  our  knowledge  must  exist. 

2.  There  is  another  mode  in  which  the  distinction  of 
the  two  elements  of  knowledge  appears,  as  I  have  already 
said :  (C.  i.  Sect.  2.)  namely  in  the  distinction  of  neces 
sary  and  contingent  or  experiential  truths.  For  of  these 
two  classes  of  truths,  the  difference  arises  from  this  ;— 
that  the  one  class  derives  its  nature  from  the  one,  and 
the  other  from  the  other,  of  the  two  elements  of  know 
ledge.  I  have  already  stated  briefly  the  difference  of 
these  two  kinds  of  truths  : — namely,  that  the  former  are 
truths  which,  we  see,  must  be  true : — the  latter  are  true, 
but  so  far  as  we  can  see,  might  be  otherwise.  The  former 
are  true  necessarily  and  universally :  the  latter  are  learnt 
from  experience  and  limited  by  experience.  Now  with 
regard  to  the  former  kind  of  truths,  I  wish  to  show  that 
the  universality  and  necessity  which  distinguish  them 
can  by  no  means  be  derived  from  experience ;  that  these 
characters  do  in  reality  flow  from  the  ideas  which  these 
truths  involve ;  and  that  when  the  necessity  of  the  truth 
is  exhibited  in  the  way  of  logical  demonstration,  it  is 
found  to  depend  upon  certain  fundamental  principles, 
(Definitions  and  Axioms,)  which  may  thus  be  considered 
as  expressing,  in  some  measure,  the  essential  characters 
of  our  ideas.  These  fundamental  principles  I  shall  after 
wards  proceed  to  discuss  and  to  exhibit  in  each  of  the 
principal  departments  of  science. 

I  shall  begin  by  considering  Necessary  Truths  more 
fully  than  I  have  yet  done.  As  I  have  already  said, 
necessary  truths  are  those  in  which  we  not  only  learn 
that  the  proposition  is  true,  but  see  that  it  must  be  true  ; 
in  which  the  negation  of  the  truth  is  not  only  false,  but 
impossible;  in  which  we  cannot,  even  by  an  effort  of 
imagination,  or  in  a  supposition,  conceive  the  reverse  of 
that  which  is  asserted. 

50  OF    IDEAS    IN    GENERAL. 

3.  That  there  are  such  truths  cannot  he  doubted. 
We  may  take,  for  example,  all  relations  of  number. 
Three  and  Two  added  together  make  Five.  We  cannot 
conceive  it  to  be  otherwise.  We  cannot,  by  any  freak 
of  thought,  imagine  Three  and  Two  to  make  Seven. 

It  may  be  said  that  this  assertion  merely  expresses 
what  we  mean  by  our  words ;  that  it  is  a  matter  of  defi 
nition  ;  that  the  proposition  is  an  identical  one. 

But  this  is  by  no  means  so.  The  definition  of  Five 
is  not  Three  and  Two,  but  Four  and  One.  How  does  it 
appear  that  Three  and  Two  is  the  same  number  as  Four 
and  One  ?  It  is  evident  that  it  is  so  ;  but  why  is  it  evi 
dent  ? — not  because  the  proposition  is  identical ;  for  if 
that  were  the  reason,  all  numerical  propositions  must  be 
evident  for  the  same  reason.  If  it  be  a  matter  of  defi 
nition  that  3  and  2  make  5,  it  must  be  a  matter  of  defi 
nition  that  39  and  27  make  G6.  But  who  will  say  that 
the  definition  of  66  is  39  and  27  ?  Yet  the  magnitude 
of  the  numbers  can  make  no  difference  in  the  ground  of 
the  truth.  How  do  we  know  that  the  product  of  13  and 
17  is  4  less  than  the  product  of  15  and  15?  We  see 
that  it  is  so,  if  we  perform  certain  operations  by  the  rules 
of  arithmetic ;  but  how  do  we  know  the  truth  of  the 
rules  of  arithmetic?  If  we  divide  123375  by  987  ac 
cording  to  the  process  taught  us  at  school,  how  are  we 
assured  that  the  result  is  correct,  and  that  the  number 
125  thus  obtained  is  really  the  number  of  times  one 
number  is  contained  in  the  other? 

The  correctness  of  the  rule,  it  may  be  replied,  can  be 
rigorously  demonstrated.  It  can  be  shewn  that  the  pro 
cess  must  inevitably  give  the  true  quotient. 

Certainly  this  can  be  shown  to  be  the  case.  And 
precisely  because  it  can  be  shown  that  the  result  must  be 
true,  we  have  here  an  example  of  a  necessary  truth ;  and 
this  truth,  it  appears,  is  not  therefore  necessary  because  it 


is  itself  evidently  identical,  however  it  may  be  possible  to 
prove  it  by  reducing  it  to  evidently  identical  propositions. 
And  the  same  is  the  case  with  all  other  numerical  propo 
sitions  ;  for,  as  we  have  said,  the  nature  of  all  of  them  is 
the  same. 

Here,  then,  we  have  instances  of  truths  which  are 
not  only  true,  but  demonstrably  and  necessarily  true. 
Now  such  truths  are,  in  this  respect  at  least,  altogether 
different  from  truths,  which,  however  certain  they  may 
be,  are  learnt  to  be  so  only  by  the  evidence  of  observa 
tion,  interpreted,  as  observation  must  be  interpreted,  by 
our  own  mental  faculties.  There  is  no  difficulty  in  find 
ing  examples  of  these  merely  observed  truths.  We  find 
that  sugar  dissolves  in  water,  and  forms  a  transparent 
fluid,  but  no  one  will  say  that  we  can  see  any  reason 
beforehand  why  the  result  must  be  so.  We  find  that  all 
animals  which  chew  the  cud  have  also  the  divided  hoof; 
but  could  any  one  have  predicted  that  this  would  be 
universally  the  case  ?  or  supposing  the  truth  of  the  rule 
to  be  known,  can  any  one  say  that  he  cannot  conceive 
the  facts  as  occurring  otherwise  ?  Water  expands  when 
it  crystallizes,  some  other  substances  contract  in  the  same 
circumstances ;  but  can  any  one  know  that  this  will  be 
so  otherwise  than  by  observation  ?  We  have  here  propo 
sitions  rigorously  true,  (we  will  assume,)  but  can  any 
one  say  they  are  necessarily  true  ?  These,  and  the  great 
mass  of  the  doctrines  established  by  induction,  are  actual, 
but  so  far  as  we  can  see,  accidental  laws ;  results  deter 
mined  by  some  unknown  selection,  not  demonstrable 
consequences  of  the  essence  of  things,  inevitable  and 
perceived  to  be  inevitable.  According  to  the  phrase 
ology  which  has  been  frequently  used  by  philosophical 
writers,  they  arc  contingent,  not  necessary  truths. 

It  is  requisite  to  insist  upon  this  opposition,  because 
no  insight    can   be    obtained    into    the    true    nature  of 

58  OF    IDEAS    IN    GENERAL. 

knowledge,  and  the  mode  of  arriving  at  it,  by  any  one 
who  does  not  clearly  appreciate  the  distinction.  The 
separation  of  truths  which  are  learnt  by  observation,  and 
truths  which  can  be  seen  to  be  true  by  a  pure  act  of 
thought,  is  one  of  the  first  and  most  essential  steps  in 
our  examination  of  the  nature  of  truth,  and  the  mode  of 
its  discovery.  If  any  one  does  not  clearly  comprehend 
this  distinction  of  necessary  and  contingent  truths,  he 
will  not  be  able  to  go  along  with  us  in  our  researches 
into  the  foundations  of  human  knowledge ;  nor,  indeed, 
to  pursue  with  success  any  speculation  on  the  subject. 
But,  in  fact,  this  distinction  is  one  that  can  hardly  fail 
to  be  at  once  understood.  It  is  insisted  upon  by  almost 
all  the  best  modern,  as  well  as  ancient,  metaphysicians"''', 
as  of  primary  importance.  And  if  any  person  does  not 
fully  apprehend,  at  first,  the  different  kinds  of  truth  thus 
pointed  out,  let  him  study,  to  some  extent,  those  sciences 
which  have  necessary  truth  for  their  subject,  as  geometry, 
or  the  properties  of  numbers,  so  as  to  obtain  a  familiar 
acquaintance  with  such  truth ;  and  he  will  then  hardly 
fail  to  see  how  different  the  evidence  of  the  propositions 
which  occur  in  these  sciences,  is  from  the  evidence  of 
the  facts  which  are  merely  learnt  from  experience. 
That  the  year  goes  through  its  course  in  365  days,  can 
only  be  known  by  observation  of  the  sun  or  stars :  that 
365  days  is  52  weeks  and  a  day,  it  requires  no  expe 
rience,  but  only  a  little  thought  to  perceive.  That  bees 
build  their  cells  in  the  form  of  hexagons,  we  cannot 
know  without  looking  at  them ;  that  regular  hexagons 
may  be  arranged  so  as  to  fill  space,  may  be  proved  with 
the  utmost  rigour,  even  if  there  were  not  in  existence 
such  a  thing  as  a  material  hexagon. 

4.     As  I  have  already  said,  one  mode  in  which  we 
may  express  the  difference  of  necessary  truths  and  truths 

*  Aristotle,  Dr.  Wliatdy,  Dugald  Stewart,  &c. 


of  experience,  is,  that  necessary  truths  are  those  of  which 
we  cannot  distinctly  conceive  the  contrary.  We  can 
very  readily  conceive  the  contrary  of  experiential  truths. 
We  can  conceive  the  stars  moving  about  the  pole  or 
across  the  sky  in  any  kind  of  curves  with  any  velocities ; 
we  can  conceive  the  moon  always  appearing  during  the 
whole  month  as  a  luminous  disk,  as  she  might  do  if  her 
light  were  inherent  and  not  borrowed.  But  we  cannot 
conceive  one  of  the  parallelograms  on  the  same  base  and 
between  the  same  parallels  larger  than  the  other ;  for 
we  find  that,  if  we  attempt  to  do  this,  when  we  separate 
the  parallelograms  into  parts,  we  have  to  conceive  one 
triangle  larger  than  another,  both  having  all  their  parts 
equal ;  which  we  cannot  conceive  at  all,  if  we  conceive 
the  triangles  distinctly.  We  make  this  impossibility 
more  clear  by  conceiving  the  triangles  to  be  placed  so 
that  two  sides  of  the  one  coincide  with  two  sides  of  the 
other ;  and  it  is  then  seen,  that  in  order  to  conceive  the 
triangles  unequal,  we  must  conceive  the  two  bases  which 
have  the  same  extremities  both  ways,  to  be  different 
lines,  though  both  straight  lines.  This  it  is  impossible 
to  conceive :  we  assent  to  the  impossibility  as  an  axiom, 
when  it  is  expressed  by  saying,  that  two  straight  lines 
cannot  inclose  a  space ;  and  thus  we  cannot  distinctly 
conceive  the  contrary  of  the  proposition  just  mentioned 
respecting  parallelograms. 

But  it  is  necessary,  in  applying  this  distinction,  to 
bear  in  mind  the  terms  of  it ; — that  we  cannot  distinct/)/ 
conceive  the  contrary  of  a  necessary  truth.  For  in  a 
certain  loose,  indistinct  way,  persons  conceive  the  con 
trary  of  necessary  geometrical  truths,  when  they  erro 
neously  conceive  false  propositions  to  be  true.  Thus, 
Hobbes  erroneously  held  that  he  had  discovered  a  means 
of  geometrically  doubling  the  cube,  as  it  is  called,  that 
is,  finding  two  mean  proportionals  between  two  given 


lines ;  a  problem  which  cannot  be  solved  by  plane 
geometry.  Hobbes  not  only  proposed  a  construction  for 
this  purpose,  but  obstinately  maintained  that  it  was 
right,  when  it  had  been  proved  to  be  wrong.  But  then, 
the  discussion  showed  how  indistinct  the  geometrical 
conceptions  of  Hobbes  were ;  for  when  his  critics  had 
proved  that  one  of  the  lines  in  his  diagram  would  not 
meet  the  other  in  the  point  which  his  reasoning  sup 
posed,  but  in  another  point  near  to  it ;  he  maintained,  in 
reply,  that  one  of  these  points  was  large  enough  to 
include  the  other,  so  that  they  might  be  considered  as 
the  same  point.  Such  a  mode  of  conceiving  the  oppo 
site  of  a  geometrical  truth,  forms  no  exception  to  the 
assertion,  that  this  opposite  cannot  be  distinctly  con 

In  like  manner,  the  indistinct  conceptions  of  children 
and  of  rude  savages  do  not  invalidate  the  distinction  cf 
necessary  and  experiential  truths.  Children  and  savages 
make  mistakes  even  with  regard  to  numbers ;  and  might 
easily  happen  to  assert  that  27  and  38  are  equal  to  G3 
or  G4.  But  such  mistakes  cannot  make  arithmetical 
truths  cease  to  be  necessary  truths.  When  any  person 
conceives  these  numbers  and  their  addition  distinctly,  by 
resolving  them  into  parts,  or  in  any  other  way,  he  sees 
that  their  sum  is  necessarily  G5.  If,  on  the  ground  of 
the  possibility  of  children  and  savages  conceiving  some 
thing  different,  it  be  held  that  this  is  not  a  necessary 
truth,  it  must  be  held  on  the  same  ground,  that  it  is  not 
a  necessary  truth  that  7  and  4  are  equal  to  11 ;  for 
children  and  savages  might  be  found  so  unfamiliar  with 
numbers  as  not  to  reject  the  assertion  that  7  and  4  are 
10,  or  even  that  4  and  3  are  G,  or  8.  But  I  suppose 
that  no  persons  would  on  such  grounds  hold  that  these 
arithmetical  truths  arc  truths  known  only  by  experi 


5.  I  have  taken  examples  of  necessary  truths  from 
the  properties  of  number  and  space;  but  such  truths  exist 
no    less   in    other   subjects,    although   the    discipline  of 
thought  which  is  requisite  to  perceive  them  distinctly, 
may  not  be  so  usual  among  men  with   regard   to   the 
sciences   of  mechanics   and   hydrostatics,  as  it  is  with 
regard  to  the  sciences  of  geometry  and  arithmetic.     Yet 
every  one  may  perceive  that  there  are  such  truths  in 
mechanics.     If  I  press  the    table    with    my    hand,   the 
table  presses  my  hand  with  an  equal  force :  here  is  a 
self-evident    and    necessary    truth.      In    any    machine, 
constructed  in  whatever   manner  to  increase  the  force 
which  I  can  exert,  it  is  certain  that  what  I  gain  in  force 
I  must  lose  in  the  velocity  which  I  communicate.     This 
is  not  a  contingent  truth,  borrowed  from  and  limited  by 
observation ;  for  a  man  of  sound  mechanical  views  applies 
it  with  like  confidence,  however  novel  be  the  construc 
tion  of  the  machine.     When  I  come  to  speak  of  the  ideas 
which  are    involved    in    our   mechanical   knowledge,   I 
may,  perhaps,  be  able  to  bring  more  clearly  into  view 
the   necessary   truth    of  general   propositions    on  such 
subjects.     That  reaction  is  equal  and  opposite  to  action, 
is  as  necessarily  true  as  that  two  straight  lines  cannot 
inclose  a  space ;  it  is  as  impossible  theoretically  to  make 
a  perpetual  motion  by  mere  mechanism  as  to  make  the 
diagonal  of  a  square  commensurable  with  the  side. 

6.  Necessary  truths  must  be  universal  truths.  If  any 
property  belong  to  a  right-angled  triangle  necessarily,  it 
must  belong  to  all  right-angled  triangles.     And  it  shall 
be  proved  in  the  following  Chapter,  that  truths  possess 
ing  these  two  characters,  of  Necessity  and  Universality, 
cannot  possibly  be  the  mere  results  of  experience. 



1.  I  HERE  employ  the  term  Experience  in  a  more  defi 
nite  and  limited  sense  than  that  which  it  possesses  in 
common  usage ;  for  I  restrict  it  to  matters  belonging  to 
the  domain  of  science.  In  such  cases,  the  knowledge 
which  we  acquire,  by  means  of  experience,  is  of  a  clear 
and  precise  nature ;  and  the  passions  and  feelings  and 
interests,  which  make  the  lessons  of  experience  in  prac 
tical  matters  so  difficult  to  read  aright,  no  longer  disturb 
and  confuse  us.  We  may,  therefore,  hope,  by  attending 
to  such  cases,  to  learn  what  efficacy  experience  really 
has,  in  the  discovery  of  truth. 

That  from  experience  (including  intentional  expe 
rience,  or  observation,}  we  obtain  much  knowledge  which 
is  highly  important,  and  which  could  not  be  procured 
from  any  other  source,  is  abundantly  clear.  We  have 
already  taken  several  examples  of  such  knowledge. 
We  know  by  experience  that  animals  which  ruminate 
are  cloven-hoofed;  and  we  know  this  in  no  other  man 
ner.  We  know,  in  like  manner,  that  all  the  planets  and 
their  satellites  revolve  round  the  sun  from  west  to  east. 
It  has  been  found  by  experience  that  all  meteoric  stones 
contain  chrome.  Many  similar  portions  of  our  know 
ledge  might  be  mentioned. 

Now  what  we  have  here  to  remark  is  this ; — that  in 
no  case  can  experience  prove  a  proposition  to  be  neces 
sarily  or  universally  true.  However  many  instances  we 
may  have  observed  of  the  truth  of  a  proposition,  yet  if  it  be 
known  merely  by  observation,  there  is  nothing  to  assure 
us  that  the  next  case  shall  not  be  an  exception  to  the  rule. 
If  it  be  strictly  true  that  every  ruminant  animal  yet 
known  has  cloven  hoofs,  we  still  cannot  be  sure  that 


some  creature  will  not  hereafter  be  discovered  which  has 
the  first  of  these  attributes  without  having  the  other. 
When  the  planets  and  their  satellites,  as  far  as  Saturn,  had 
been  all  found  to  move  round  the  sun  in  one  direction, 
it  was  still  possible  that  there  might  be  other  such  bodies 
not  obeying  this  rule ;  and,  accordingly,  when  the  satel 
lites  of  Uranus  were  detected,  they  appeared  to  offer  an 
exception  of  this  kind.  Even  in  the  mathematical  sciences, 
we  have  examples  of  such  rules  suggested  by  experience, 
and  also  of  their  precariousness.  However  far  they  may 
have  been  tested,  we  cannot  depend  upon  their  correct 
ness,  except  we  see  some  reason  for  the  rule.  For 
instance,  various  rules  have  been  given,  for  the  purpose 
of  pointing  out  prime  numbers;  that  is,  those  which  can 
not  be  divided  by  any  other  number.  We  may  try,  as 
an  example  of  such  a  rule,  this  one — any  odd  power  of 
the  number  two,  diminished  by  one.  Thus  the  third 
power  of  two,  diminished  by  one,  is  seven;  the  fifth 
power,  diminished  by  one,  is  thirty-one ;  the  seventh 
power  so  diminished  is  one  hundred  and  twenty-seven. 
All  these  are  prime  numbers :  and  we  might  be  led  to 
suppose  that  the  rule  is  universal.  But  the  next  ex 
ample  shows  us  the  fallaciousness  of  such  a  belief.  The 
ninth  power  of  two,  diminished  by  one,  is  five  hundred 
and  eleven,  which  is  not  a  prime,  being  divisible  by  seven. 
Experience  must  always  consist  of  a  limited  number 
of  observations.  And,  however  numerous  these  may  be, 
they  can  show  nothing  with  regard  to  the  infinite 
number  of  cases  in  which  the  experiment  has  not  been 
made.  Experience  being  thus  unable  to  prove  a  fact 
to  be  universal,  is,  as  will  readily  be  seen,  still  more 
incapable  of  proving  a  truth  to  be  necessary.  Expe 
rience  cannot,  indeed,  offer  the  smallest  ground  for  the 
necessity  of  a  proposition.  She  can  observe  and  record 
what  has  happened ;  but  she  cannot  find,  in  any  case,  or 

04  OF    IDEAS    IN    GENERAL. 

in  any  accumulation  of  cases,  any  reason  for  what  mutt 
happen.  She  may  see  objects  side  by  side ;  but  she 
cannot  see  a  reason  why  they  must  ever  be  side  by  side. 
She  finds  certain  events  to  occur  in  succession ;  but  the 
succession  supplies,  in  its  occurrence,  no  reason  for  its 
recurrence.  She  contemplates  external  objects  ;  but  she 
cannot  detect  any  internal  bond,  which  indissolubly 
connects  the  future  with  the  past,  the  possible  with  the 
real.  To  learn  a  proposition  by  experience,  and  to  see 
it  to  be  necessarily  true,  are  two  altogether  different  pro 
cesses  of  thought. 

2.  But  it  may  be  said,  that  we  do  learn  by  means 
of  observation  and  experience  many  universal  truths ; 
indeed,  all  the  general  truths  of  which  science  consists. 
Is  not  the  doctrine  of  universal  gravitation  learnt  by 
experience  ?  Are  not  the  laws  of  motion,  the  properties 
of  light,  the  general  principles  of  chemistry,  so  learnt  ? 
How,  with  these  examples  before  us,  can  we  say  that 
experience  teaches  no  universal  truths? 

To  this  we  reply,  that  these  truths  can  only  be 
known  to  be  general,  not  universal,  if  they  depend  upon 
experience  alone.  Experience  cannot  bestow  that  uni 
versality  which  she  herself  cannot  have,  and  that  necessity 
of  which  she  has  no  comprehension.  If  these  doctrines 
are  universally  true,  this  universality  flows  from  the  ideas 
which  we  apply  to  our  experience,  and  which  are,  as  we 
have  seen,  the  real  sources  of  necessary  truth.  How  far 
these  ideas  can  communicate  their  universality  and 
necessity  to  the  results  of  experience,  it  will  hereafter 
be  our  business  to  consider.  It  will  then  appear,  that 
when  the  mind  collects  from  observation  truths  of  a  wide 
and  comprehensive  kind,  which  approach  to  the  sim 
plicity  and  universality  of  the  truths  of  pure  science ; 
she  gives  them  this  character  by  throwing  upon  them 
the  light  of  her  own  Fundamental  Ideas. 


But  the  truths  which  we  discover  by  observation  of 
the   external  world,  even  when  most,  strikingly  simple 
and  universal,  are  not  necessary  truths.     Is  the  doctrine 
of  universal  gravitation  necessarily  true  ?    It  was  doubted 
by  Clairaut  (so  far  as  it  refers  to  the  moon),  when  the 
progression  of  the  apogee  in  fact  appeared  to  be  twice 
as  great  as  the  theory  admitted.     It  has  been  doubted, 
even  more  recently,  with  respect  to  the  planets,  their 
mutual  perturbations  appearing  to  indicate  a  deviation 
from  the  law.     It  is  doubted  still,  by  some  persons,  with 
respect  to   the    double   stars.     But   suppose   all   these 
doubts  to  be  banished,  and  the  law  to  be  universal ;  is  it 
then  proved  to  be  necessary  ?     Manifestly  not :  the  very 
existence  of  these  doubts  proves  that  it  is  not  so.     For 
the  doubts  were  dissipated  by  reference  to  observation 
and  calculation,  not  by  reasoning  on  the  nature  of  the 
law.     Clairaut's  difficulty  was  removed  by  a  more  exact 
calculation  of  the  effect  of  the  sun's  force  on  the  motion 
of  the  apogee.     The  suggestion  of  Bessel,  that  the  in 
tensity   of  gravitation   might  be  different  for  different 
planets,  was  found  to  be  unnecessary,  when  Professor 
Airy  gave  a  more  accurate  determination  of  the  mass  of 
Jupiter.     And  the  question  whether  the  extension  of  the 
law  of  the  inverse  square  to  the  double  stars  be  true, 
(one  of  the v  most  remarkable  questions  now  before  the 
scientific  world,)  must  be  answered,  not  by  any  specula 
tions  concerning  what  the  laws  of  attraction  must  neces 
sarily  be,  but  by  carefully  determining  the  actual  laws 
of  the  motion  of  these  curious  objects,  by  means  of  the 
observations  such  as  those  which  Sir  John  Hcrschel  has 
collected  for  that  purpose,  by  his  unexampled  survey  of 
both  hemispheres  of  the  sky.     And  since  the  extent  of 
this  truth  is  thus  to  be  determined  by  reference  to  ob 
served  facts,  it  is  clear  that  no  mere  accumulation  of 
VOL.  i.    w.  r.  F 

06  OF    IDEAS    IN    GENERAL. 

them  can  make  its  universality  certain,  or  its  necessity 

Thus  no  knowledge  of  the  necessity  of  any  truths 
can  result  from  the  observation  of  what  really  happens. 
This  being  clearly  understood,  we  are  led  to  an  import 
ant  inquiry. 

The  characters  of  universality  and  necessity  in  the 
truths  which  form  part  of  our  knowledge,  can  never 
be  derived  from  experience,  by  which  so  large  a  part 
of  our  knowledge  is  obtained.  But  since,  as  we  have 
seen,  we  really  do  possess  a  large  body  of  truths  which 
are  necessary,  and  because  necessary,  therefore  universal, 
the  question  still  recurs,  from  what  source  these  charac 
ters  of  universality  and  necessity  are  derived. 

The  answer  to  this  question  we  will  attempt  to  give 
in  the  next  chapter. 


1 .  To  the  question  just  stated,  I  reply,  that  the  neces 
sity  and  universality  of  the  truths  which  form  a  part  of 
our  knowledge,  are  derived  from  the  Fundamental  Ideas 
which  those  truths  involve.  These  ideas  entirely  shape 
and  circumscribe  our  knowledge ;  they  regulate  the  ac 
tive  operations  of  our  minds,  without  which  our  passive 
sensations  do  not  become  knowledge.  They  govern 
these  operations,  according  to  rules  which  are  not  only 
fixed  and  permanent,  but  which  may  be  expressed  in 
plain  and  definite  terms;  and  these  rules,  when  thus 
expressed,  may  be  made  the  basis  of  demonstrations  by 
which  the  necessary  relations  imparted  to  our  know 
ledge  by  our  Ideas  may  be  traced  to  their  consequences 
in  the  most  remote  ramifications  of  scientific  truth. 


These  enunciations  of  the  necessary  and  evident  con 
ditions  imposed  upon  our  knowledge  by  the  Fundamental 
Ideas  which  it  involves,  are  termed  Axioms.  Thus  the 
Axioms  of  Geometry  express  the  necessary  conditions 
which  result  from  the  Idea  of  Space ;  the  Axioms  of 
Mechanics  express  the  necessary  conditions  which  flow 
from  the  Ideas  of  Force  and  Motion  ;  and  so  on. 

2.  It  will  be  the  office  of  several  of  the  succeeding 
Books  of  this  work  to  establish  and  illustrate  in  detail 
what  I  have  thus  stated  in  general  terms.    I  shall  there 
pass  in  review  many  of  the  most  important  fundamental 
ideas  on  which  the  existing  body  of  our  science  depends ; 
and  I  shall  endeavour  to  show,  for  each  such  idea  in 
succession,  that  knowledge  involves  an  active  as  well  as 
a  passive  element ;  that  it  is  not  possible  without  an  act 
of  the  mind,  regulated  by  certain  laws.     I  shall  further 
attempt  to  enumerate  some  of  the  principal  fundamental 
relations   which   each   idea   thus    introduces    into    our 
thoughts,  and  to  express  them  by  means  of  definitions 
and  axioms,  and  other  suitable  forms. 

I  will  only  add  a  remark  or  two  to  illustrate  further 
this  view  of  the  ideal  grounds  of  our  knowledge. 

3.  To  persons  familiar  with  any  of  the  demonstrative 
sciences,  it  will  be  apparent  that  if  we  state  all  the 
Definitions   and  Axioms   which   are    employed   in   the 
demonstrations,  we  state  the  whole  basis  on  which  those 
reasonings  rest.    For  the  whole  process  of  demonstrative 
or  deductive  reasoning  in  any  science,  (as  in  geometry, 
for  instance,)  consists  entirely  in  combining  some  of  these 
first  principles  so  as  to  obtain  the  simplest  propositions 
of. the  science;  then  combining  these  so  as  to  obtain 
other  propositions  of  greater  complexity;  and  so  on,  till 
we  advance  to  the  most  recondite  demonstrable  truths ; 
these  last,  however,  intricate  and  unexpected,  still  in 
volving  no  principles  except  the  original  definitions  and 

F  2 

68  OF    IDEAS    IN    GENERAL. 

axioms.  Thus,  by  combining  the  Definition  of  a  triangle, 
and  the  Definitions  of  equal  lines  and  equal  angles, 
namely,  that  they  are  such  as  when  applied  to  each 
other,  coincide,  with  the  Axiom  respecting  straight  lines 
(that  two  such  lines  cannot  inclose  a  space,)  we  demon 
strate  the  equality  of  triangles,  under  certain  assumed 
conditions.  Again,  by  combining  this  result  with  the 
Definition  of  parallelograms,  and  with  the  Axiom  that  if 
cquals  be  taken  from  equals  the  wholes-  are  equal,  we 
prove  the  equality  of  parallelograms  between  the  same 
parallels  and  upon  the  same  base.  From  this  proposi 
tion,  again,  we  prove  the  equality  of  the  square  on  the 
hypotenuse  of  a  triangle  to  the  squares  on  the  two  sides 
containing  the  right  angle.  But  in  all  this  there  is 
nothing  contained  which  is  not  rigorously  the  result  of 
our  geometrical  Definitions  and  Axioms.  All  the  rest 
of  our  treatises  of  geometry  consists  only  of  terms  and 
phrases  of  reasoning,  the  object  of  which  is  to  connect 
those  first  principles,  and  to  exhibit  the  effects  of  their 
combination  in  the  shape  of  demonstration. 

4.  This  combination  of  first  principles  takes  place 
according  to  the  forms  and  rules  of  Logic.  All  the 
steps  of  the  demonstration  may  be  stated  in  the  shape  in 
which  logicians  are  accustomed  to  exhibit  processes  of 
reasoning  in  order  to  show  their  conclusiveness,  that  is, 
in  Syllogisms.  Thus  our  geometrical  reasonings  might 
be  resolved  into  such  steps  as  the  following : — 

All  straight  lines  drawn  from  the  centre  of  a  circle 
to  its  circumference  are  equal : 

But  the  straight  lines  AB,  AC,  are  drawn  from  the 
centre  of  a  circle  to  its  circumference  :  . 

Therefore  the  straignt  lines  AB,  AC,  are  equal. 

Each  step  of  geometrical,  and  all  other  demonstra 
tive  reasoning,  may  be  resolved  into  three  such  clauses 
as  these  ;  and  these  three  clauses  are  termed  respectively, 


the  major  premiss,  the  minor  premiss,  and  the  conclu 
sion;  or,  more  briefly,  the  major,  the  minor,  and  the 

The  principle  which  justifies  the  reasoning  when 
exhibited  in  this  syllogistic  form,  is  this : — that  a  truth 
which  can  be  asserted  as  generally,  or  rather  as  univer 
sally  true,  can  be  asserted  as  true  also  in  each  particular 
case.  The  minor  only  asserts  a  certain  particular  case 
to  be  an  example  of  such  conditions  as  are  spoken  of  in 
the  major;  and  hence  the  conclusion,  which  is  true  of 
the  major  by  supposition,  is  true  of  the  minor  by  conse 
quence  ;  and  thus  we  proceed  from  syllogism  to  syl 
logism,  in  each  one  employing  some  general  truth  in 
some  particular  instance.  Any  proof  which  occurs  in 
geometry,  or  any  other  science  of  demonstration,  may 
thus  be  reduced  to  a  series  of  processes,  in  each  of 
which  we  pass  from  some  general  proposition  to  the 
narrower  and  more  special  propositions  which  it  in 
cludes.  And  this  process  of  deriving  truths  by  the  mere 
combination  of  general  principles,  applied  in  particular 
hypothetical  cases,  is  called  deduction;  being  opposed 
to  induction,  in  which,  as  we  have  seen,  (Chap.  I.  Sect.  3.) 
a  new  general  principle  is  introduced  at  every  step. 

5.  Now  we  have  to  remark  that,  this  being  so,  how 
ever  far  we  follow  such  deductive  reasoning,  we  can 
never  have,  in  our  conclusion  any  truth  which  is  not 
virtually  included  in  the  original  principles  from  which 
the  reasoning  started.  For  since  at  any  step  we  merely 
take  out  of  a  general  proposition  something  included  in 
it,  while  at  the  preceding  step  we  have  taken  this  ge 
neral  proposition  out  of  one  more  general,  and  so  on 
perpetually,  it  is  manifest  that  our  last  result  was  really 
included  in  the  principle  or  principles  with  which  we 
began.  I  say  principles,  because,  although  our  logical 
conclusion  can  only  exhibit  the  legitimate  issue  of  our 

70  OF    IDEAS   IN    GENEIIAL. 

first  principles,  it  may,  nevertheless,  contain  the  result 
of  the  combination  of  several  such  principles,  and  may 
thus  assume  a  great  degree  of  complexity,  and  may  ap 
pear  so  far  removed  from  the  parent  truths,  as  to  betray 
at  first  sight  hardly  any  relationship  with  them.  Thus 
the  proposition  which  has  already  been  quoted  respect 
ing  the  squares  oii  the  sides  of  a  right-angled  triangle, 
contains  the  results  of  many  elementary  principles ;  as, 
the  definitions  of  parallels,  triangle,  and  square ;  the 
axioms  respecting  straight  lines,  and  respecting  paral 
lels;  and,  perhaps,  others.  The  conclusion  is  compli 
cated  by  containing  the  effects  of  the  combination  of  all 
these  elements ;  but  it  contains  nothing,  and  can  contain 
nothing,  but  such  elements  and  their  combinations. 

This  doctrine,  that  logical  reasoning  produces  no  new 
truths,  but  only  unfolds  and  brings  into  view  those  truths 
which  were,  in  effect,  contained  in  the  first  principles  of 
the  reasoning,  is  assented  to  by  almost  all  who,  in 
modern  times,  have  attended  to  the  science  of  logic. 
Such  a  view  is  admitted  both  by  those  who  defend,  and 
by  those  who  depreciate  the  value  of  logic.  "  Whatever 
is  established  by  reasoning,  must  have  been  contained 
and  virtually  asserted  in  the  premises"'."  "The  only 
truth  which  such  propositions  can  possess  consists  in 
conformity  to  the  original  principles." 

In  this  manner  the  whole  substance  of  our  geometry 
is  reduced  to  the  Definitions  and  Axioms  which  we 
employ  in  our  elementary  reasonings ;  and  in  like  man 
ner  we  reduce  the  demonstrative  truths  of  any  other 
science  to  the  definitions  and  axioms  which  we  there 

6.  But  in  reference  to  this  subject,  it  has  sometimes 
been  said  that  demonstrative  sciences  do  in  reality  depend 
upon  Definitions  only;  and  that  no  additional  kind  of 

*  Whatelcy's  Logic,  pp.  237,  238. 


principle,  such  as  we  have  supposed  Axioms  to  be,  is 
absolutely  required.  It  has  been  asserted  that  in  geo 
metry,  for  example,  the  source  of  the  necessary  truth  of 
our  propositions  is  this,  that  they  depend  upon  definitions 
alone,  and  consequently  merely  state  the  identity  of  the 
same  thing  under  different  aspects. 

That  in  the  sciences  which  admit  of  demonstration, 
as  geometry,  mechanics,  and  the  like,  Axioms  as  well  as 
Definitions  are  needed,  in  order  to  express  the  grounds 
of  our  necessary  convictions,  must  be  shown  hereafter 
by  an  examination  of  each  of  these  sciences  in  particular. 
But  that  the  propositions  of  these  sciences,  those  of  geo 
metry  for  example,  do  not  merely  assert  the  identity  of 
the  same  thing,  will,  I  think,  be  generally  allowed,  if  we 
consider  the  assertions  which  we  are  enabled  to  make. 
When  we  declare  that  "  a  straight  line  is  the  shortest 
distance  between  two  points,"  is  this  merely  an  identical 
proposition?  the  definition  of  a  straight  line  in  another 
form  ?  Not  so  :  the  definition  of  a  straight  line  involves 
the  notion  of  form  only,  and  does  not  contain  anything 
about  magnitude ;  consequently,  it  cannot  contain  any 
thing  equivalent  to  "  shortest."  Thus  the  propositions 
of  geometry  are  not  merely  identical  propositions ;  nor 
have  we  in  their  general  character  anything  to  coun 
tenance  the  assertion,  that  they  are  the  results  of  defi 
nitions  alone.  And  when  we  come  to  examine  this  and 
other  sciences  more  closely,  we  shall  find  that  axioms, 
such  as  are  usually  in  our  treatises  made  the  funda 
mental  principles  of  our  demonstrations,  neither  have 
ever  been,  nor  can  be,  dispensed  with.  Axioms,  as  well 
as  Definitions,  are  in  all  cases  requisite,  in  order  pro 
perly  to  exhibit  the  grounds  of  necessary  truth. 

7.  Thus  the  real  logical  basis  of  every  body  of  demon 
strated  truths  are  the  Definitions  and  Axioms  which  are 
the  first  principles  of  the  reasonings.  But  when  we  arc 

72  OF   IDEAS    IX    GENERAL. 

arrived  at  this  point,  the  question  further  occurs,  what 
is  the  ground  of  the  truth  of  these  Axioms?  It  is  not 
the  logical,  but  the  philosophical,  not  the  formal,  but  the 
real  foundation  of  necessary  truth,  which  we  are  seeking. 
Hence  this  inquiry  necessarily  comes  before  us,  What 
is  the  ground  of  the  Axioms  of  Geometry,  of  Mechanics, 
and  of  any  other  demonstrable  science  ? 

The  answer  which  we  are  led  to  give,  by  the  view 
which  we  have  taken  of  the  nature  of  knowledge,  has 
already  been  stated.  The  ground  of  the  axioms  belong 
ing  to  each  science  is  the  Idea  which  the  axiom  involves. 
The  ground  of  the  Axioms  of  Geometry  is  the  Idea  of 
Space:  the  ground  of  the  Axioms  of  Mechanics  is  the 
Idea  of  Force,  of  Action  and  Reaction,  and  the  like.  And 
hence  these  Ideas  are  Fundamental  Ideas ;  and  since  they 
are  thus  the  foundations,  not  only  of  demonstration  but 
of  truth,  an  examination  into  their  real  import  and 
nature  is  of  the  greatest  consequence  to  our  purpose. 

8.  Not  only  the  Axioms,  but  the  Definitions  which 
form  the  basis  of  our  reasonings,  depend  upon  our  Fun 
damental  Ideas.  And  the  Definitions  are  not  arbitrary 
definitions,  but  are  determined  by  a  necessity  no  less 
rigorous  than  the  Axioms  themselves.  We  could  not 
think  of  geometrical  truths  without  conceiving  a  circle ; 
and  we  could  not  reason  concerning  such  truths  without 
defining  a  circle  in  some  mode  equivalent  to  that  which 
is  commonly  adopted.  The  Definitions  of  parallels,  of 
right  angles,  and  the  like,  are  quite  as  necessarily  pre 
scribed  by  the  nature  of  the  case,  as  the  Axioms  which 
these  Definitions  bring  with  them.  Indeed  we  may 
substitute  one  of  these  kinds  of  principles  for  another. 
We  cannot  always  put  a  Definition  in  the  place  of  an 
Axiom ;  but  we  may  always  find  an  Axiom  which  shall 
take  the  place  of  a  Definition.  If  we  assume  a  proper 
Axiom  respecting  straight  lines,  we  need  no  Definition 


of  a  straight  line.  But  in  whatever  shape  the  principle 
appear,  as  Definition  or  as  Axiom,  it  has  about  it  nothing 
casual  or  arbitrary,  but  is  determined  to  be  what  it  is,  as 
to  its  import,  by  the  most  rigorous  necessity,  growing 
out  of  the  Idea  of  Space. 

9.  These  principles, — Definitions,  and  Axioms, — thus 
exhibiting  the  primary  developements  of  a  fundamental 
idea,  do  in  fact  express  the  idea,  so  far  as  its  expression 
in  words  forms  part  of  our  science.  They  are  different 
views  of  the  same  body  of  truth ;  and  though  each  prin 
ciple,  by  itself,  exhibits  only  one  aspect  of  this  body, 
taken  together  they  convey  a  sufficient  conception  of  it 
for  our  purposes.  The  Idea  itself  cannot  be  fixed  in 
words ;  but  these  various  lines  of  truth  proceeding  from 
it,  suggest  sufficiently  to  a  fitly-prepared  mind,  the  place 
where  the  idea  resides,  its  nature,  and  its  efficacy. 

It  is  true  that  these  principles, — our  elementary  Defi 
nitions  and  Axioms, — even  taken  altogether,  express  the 
Idea  incompletely.  Thus  the  Definitions  and  Axioms  of 
Geometry,  as  they  are  stated  in  our  elementary  works, 
do  not  fully  express  the  Idea  of  Space  as  it  exists  in  our 
minds.  For,  in  addition  to  these,  other  Axioms,  inde 
pendent  of  these,  and  no  less  evident,  can  be  stated ;  and 
are  in  fact  stated  when  we  come  to  the  Higher  Geo 
metry.  Such,  for  instance,  is  the  Axiom  of  Archimedes 
—that  a  curve  line  which  joins  two  points  is  less  than  a 
broken  line  which  joins  the  same  points  and  includes  the 
curve.  And  thus  the  Idea  is  disclosed  but  not  fully  re 
vealed,  imparted  but  not  transfused,  by  the  use  we  make 
of  it  in  science.  When  we  have  taken  from  the  fountain 
so  much  as  serves  our  purpose,  there  still  remains  behind 
a  deep  well  of  truth,  which  we  have  not  exhausted,  and 
which  we  may  easily  believe  to  be  inexhaustible. 



1.  BY  the  course  of  speculation  contained  in  the  last 
three  Chapters,  we  are  again  led  to  the  conclusion  which 
we  have  already  stated,  that  our  knowledge  contains  an 
ideal  element,  and  that  this  element  is  not  derived  from 
experience.  For  we  have  seen  that  there  are  proposi 
tions  which  are  known  to  be  necessarily  true ;  and  that 
such  knowledge  is  not,  and  cannot  be,  obtained  by  mere 
observation  of  actual  facts.  It  has  been  shown,  also, 
that  these  necessary  truths  are  the  results  of  certain  fun 
damental  ideas,  such  as  those  of  space,  number,  and  the 
like.  Hence  it  follows  inevitably  that  these  ideas  and 
others  of  the  same  kind  are  not  derived  from  experience. 
For  these  ideas  possess  a  power  of  infusing  into  their 
developements  that  very  necessity  which  experience  can 
in  no  way  bestow.  This  power  they  do  not  borrow  from 
the  external  world,  but  possess  by  their  own  nature. 
Thus  we  unfold  out  of  the  Idea  of  Space  the  propositions 
of  geometry,  which  are  plainly  truths  of  the  most  rigor 
ous  necessity  and  universality.  But  if  the  idea  of  space 
were  merely  collected  from  observation  of  the  external 
world,  it  could  never  enable  or  entitle  us  to  assert  such 
propositions :  it  could  never  authorize  us  to  say  that  not 
merely  some  lines,  but  all  lines,  not  only  have,  but  must 
have,  those  properties  which  geometry  teaches.  Geo 
metry  in  every  proposition  speaks  a  language  which 
experience  never  dares  to  utter;  and  indeed  of  which 
she  but  half  comprehends  the  meaning.  Experience 
sees  that  the  assertions  are  true,  but  she  sees'  not  how 
profound  and  absolute  is  their  truth.  She  unhesitatingly 
assents  to  the  laws  which  geometry  delivers,  but  she  does 


not  pretend  to  see  the  origin  of  their  obligation.  She 
is  always  ready  to  acknowledge  the  sway  of  pure  scien 
tific  principles  as  a  matter  of  fact,  but  she  does  not 
dream  of  offering  her  opinion  on  their  authority  as  a 
matter  of  right;  still  less  can  she  justly  claim  to  be  her 
self  the  source  of  that  authority. 

David  Hume  asserted'-,  that  we  are  incapable  of 
seeing  in  any  of  the  appearances  which  the  world  pre 
sents  anything  of  necessary  connexion ;  and  hence  he 
inferred  that  our  knowledge  cannot  extend  to  any  such 
connexion.  It  will  be  seen  from  what  we  have  said  that 
we  assent  to  his  remark  as  to  the  fact,  but  we  differ  from 
him  altogether  in  the  consequence  to  be  drawn  from  it. 
Our  inference  from  Hume's  observation  is,  not  the  truth 
of  his  conclusion,  but  the  falsehood  of  his  premises  ;— 
not  that,  therefore,  we  can  know  nothing  of  natural  con 
nexion,  but  that,  therefore,  we  have  some  other  source  of 
knowledge  than  experience : — not,  that  we  can  have  no 
idea  of  connexion  or  causation,  because,  in  his  language, 
it  cannot  be  the  copy  of  an  impression ;  but  that  since 
we  have  such  an  idea,  our  ideas  are  not  the  copies  of 
our  impressions. 

Since  it  thus  appears  that  our  fundamental  ideas  are 
not  acquired  from  the  external  world  by  our  senses,  but 
have  some  separate  and  independent  origin,  it  is  im 
portant  for  us  to  examine  their  nature  and  properties,  as 
they  exist  in  themselves;  and  this  it  will  be  our  business 
to  do  through  a  portion  of  the  following  pages.  But  it 
may  be  proper  first  to  notice  one  or  two  objections 
which  may  possibly  occur  to  some  readers. 

2.  It  may  be  said  that  without  the  use  of  our  senses, 
of  sight  and  touch,  for  instance,  we  should  never  have 
any  idea  of  space ;  that  this  idea,  therefore,  may  properly 
be  said  to  be  derived  from  those  senses.  And  to  this  I 

*  Essays,  Vol.  u.  p.  70. 

76  OF    IDEAS    IN    GENERAL. 

reply,  by  referring  to  a  parallel  instance.  Without  light 
we  should  have  no  perception  of  visible  figure ;  yet  the 
power  of  perceiving  visible  figure  cannot  be  said  to  be 
derived  from  the  light,  but  resides  in  the  structure  of  the 
eye.  If  we  had  never  seen  objects  in  the  light,  we 
should  be  quite  unaware  that  we  possessed  a  power  of 
vision ;  yet  we  should  not  possess  it  the  less  on  that 
account.  If  we  had  never  exercised  the  senses  of  sight 
and  touch  (if  we  can  conceive  such  a  state  of  human  ex 
istence)  we  know  not  that  we  should  be  conscious  of  an 
idea  of  space.  But  the  light  reveals  to  us  at  the  same 
time  the  existence  of  external  objects  and  our  own  power 
of  seeing.  And  in  a  very  similar  -manner,  the  exercise 
of  our  senses  discloses  to  us,  at  the  same  time,  the  ex 
ternal  world,  and  our  own  ideas  of  space,  time,  and  other 
conditions,  without  which  the  external  world  can  neither 
be  observed  nor  conceived.  That  light  is  necessary  to 
vision,  does  not,  in  any  degree,  supersede  the  importance 
of  a  separate  examination  of  the  laws  of  our  visual 
powers,  if  we  would  understand  the  nature  of  our  own 
bodily  faculties  and  the  extent  of  the  information  they 
can  give  us.  In  like  manner,  the  fact  that  intercourse 
with  the  external  world  is  necessary  for  the  conscious 
employment  of  our  ideas,  does  not  make  it  the  less  es 
sential  for  us  to  examine  those  ideas  in  their  most  inti 
mate  structure,  in  order  that  \ve  may  understand  the 
grounds  and  limits  of  our  knowledge.  Even  before  we 
see  a  single  object,  we  have  a  faculty  of  vision ;  and  in 
like  manner,  if  we  can  suppose  a  man  who  has  never 
contemplated  an  object  in  space  or  time,  we  must  still 
assume  him  to  have  the  faculties  of  entertaining  the  ideas 
of  space  and  time,  which  faculties  are  called  into  play 
on  the  very  first  occasion  of  the  use  of  the  senses. 

3.  In  answer  to  such  remarks  as  the  above,  it  has 
sometimes  been  said  that  to  assume  separate  faculties  in 


the  mind  for  so  many  different  processes  of  thought,  is  to 
give  a  mere  verbal  explanation,  sinee  we  learn  nothing 
concerning  our  idea  of  space  by  being  told  that  we  have 
a  faculty  of  forming  such  an  idea.  It  has  been  said  that 
this  course  of  explanation  leads  to  an  endless  multipli 
cation  of  elements  in  man's  nature,  without  any  advan 
tage  to  our  knowledge  of  his  true  constitution.  We 
may,  it  is  said,  assert  man  to  have  a  faculty  of  walking, 
of  standing,  of  breathing,  of  speaking ;  but  what,  it  is 
asked,  is  gained  by  such  assertions?  To  this  I  reply,  that 
we  undoubtedly  have  such  faculties  as  those  just  named; 
that  it  is  by  no  means  unimportant  to  consider  them;  and 
that  the  main  question  in  such  cases  is,  whether  they  are 
separate  and  independent  faculties,  or  complex  and  deri 
vative  ones ;  and,  if  the  latter  be  the  case,  what  are  the 
simple  and  original  faculties  by  the  combination  of  which 
the  others  are  produced.  In  walking,  standing,  breath 
ing,  for  instance,  a  great  part  of  the  operation  can  be 
reduced  to  one  single  faculty ;  the  voluntary  exercise  of 
our  muscles.  But  in  breathing  this  does  not  appear  to 
be  the  whole  of  the  process.  The  operation  is,  in  part  at 
least,  involuntary ;  and  it  has  been  held  that  there  is  a 
certain  sympathetic  action  of  the  nerves,  in  addition  to 
the  voluntary  agency  which  they  transmit,  which  is  essen 
tial  to  the  function.  To  determine  whether  or  no  this 
sympathetic  faculty  is  real  and  distinct,  and  if  so,  what 
are  its  laws  and  limits,  is  certainly  a  highly  philosophical 
inquiry,  and  well  deserving  the  attention  which  has  been 
bestowed  upon  it  by  eminent  physiologists.  And  just  of 
the  same  nature  are  the  inquiries  with  respect  to  man's 
intellectual  constitution,  on  which  we  propose  to  enter. 
For  instance,  man  has  a  faculty  of  apprehending  time, 
and  a  faculty  of  reckoning  numbers:  arc  these  distinct,  or 
is  one  faculty  derived  from  the  other?  To  analy/e  the 
various  combinations  of  our  ideas  and  observations  into 

78  OF    IDEAS    IN    GENERAL. 

the  original  faculties  which  they  involve ;  to  show  that 
these  faculties  are  original,  and  not  capable  of  further 
analysis :  to  point  out  the  characters  which  mark  these 
faculties  and  lead  to  the  most  important  features  of  our 
knowledge; — these  are  the  kind  of  researches  on  which 
we  have  now  to  enter,  and  these,  we  trust,  will  be  found 
to  be  far  from  idle  or  useless  parts  of  our  plan.  If  we 
succeed  in  such  attempts,  it  will  appear  that  it  is  by 
no  means  a  frivolous  or  superfluous  step  to  distinguish 
separate  faculties  in  the  mind.  If  we  do  not  learn  much 
by  being  told  that  we  have  a  faculty  of  forming  the  idea 
of  space,  wre  at  least,  by  such  a  commencement,  circum 
scribe  a  certain  portion  of  the  field  of  our  investigations, 
which,  we  shall  afterwards  endeavour  to  show,  requires 
and  rewards  a  special  examination.  And  though  we  shall 
thus  have  to  separate  the  domain  of  our  philosophy  into 
many  provinces,  these  are,  as  we  trust  it  will  appear, 
neither  arbitrarily  assigned,  nor  vague  in  their  limits, 
nor  infinite  in  number. 


WE  proceed,  in  the  ensuing  Books,  to  the  closer  exami 
nation  of  a  considerable  number  of  those  Fundamental 
Ideas  on  which  the  sciences,  hitherto  most  successfully 
cultivated,  are  founded.  In  this  task,  our  objects  will 
be  to  explain  and  analyze  such  Ideas  so  as  to  bring  into 
view  the  Definitions  and  Axioms,  or  other  forms,  in 
which  we  may  clothe  the  conditions  to  which  our  specu 
lative  knowledge  is  subjected.  I  shall  also  try  to  prove, 
for  some  of  these  Ideas  in  particular,  what  has  been 
already  urged  respecting  them  in  general,  that  they  are 


not  derived  from  observation,  but  necessarily  impose 
their  conditions  upon  that  knowledge  of  which  observa 
tion  supplies  the  materials.  I  shall  further,  in  some 
cases,  endeavour  to  trace  the  history  of  these  Ideas  as 
they  have  successively  come  into  notice  in  the  progress 
of  science;  the  gradual  developement  by  which  they  have 
arrived  at  their  due  purity  and  clearness;  and,  as  a 
necessary  part  of  such  a  history,  I  shall  give  a  view  of 
some  of  the  principal  controversies  which  have  taken 
place  with  regard  to  each  portion  of  knowledge. 

An  exposition  and  discussion  of  the  Fundamental 
Ideas  of  each  Science  may,  with  great  propriety,  be 
termed  the  PHILOSOPHY  OF  such  SCIENCE.  These  ideas 
contain  in  themselves  the  elements  of  those  truths  which 
the  science  discovers  and  enunciates;  and  in  the  progress 
of  the  sciences,  both  in  the  world  at  large  and  in  the 
mind  of  each  individual  student,  the  most  important 
steps  consist  in  apprehending  these  ideas  clearly,  and  in 
bringing  them  into  accordance  with  the  observed  facts. 
I  shall,  therefore,  in  a  series  of  Books,  treat  of  the  Phi 
losophy  of  the  Pure  Sciences,  the  Philosophy  of  the 
Mechanical  Sciences,  the  Philosophy  of  Chemistry,  and 
the  like,  and  shall  analyze  and  examine  the  ideas  which 
these  sciences  respectively  involve. 

In  this  undertaking,  inevitably  somewhat  long,  and 
involving  many  deep  and  subtle  discussions,  I  shall  take, 
as  a  chart  of  the  country  before  me,  by  which  my  course 
is  to  be  guided,  the  scheme  of  the  sciences  which  I  was 
led  to  form  by  travelling  over  the  history  of  each  in 
order*.  Each  of  the  sciences  of  which  I  then  narrated 
the  progress,  depends  upon  several  of  the  Fundamental 
Ideas  of  which  I  have  to  speak :  some  of  these  Ideas  are 
peculiar  to  one  field  of  speculation,  others  are  common 
to  more.  A  previous  enumeration  of  Ideas  thus  collected 

*  History  of  lite  Inductive  Sciences. 

80  OF    IDEAS    IN    GENERAL. 

may  serve  both  to  show  the  course  and  limits  of  this  part 
of  our  plan,  and  the  variety  of  interest  which  it  offers. 

I  shall,  then,  successively,  have  to  speak  of  the  Ideas 
which  are  the  foundation  of  Geometry  and  Arithmetic, 
(and  which  also  regulate  all  sciences  depending  upon 
these,  as  Astronomy  and  Mechanics;)  namely,  the  Ideas 
of  Space,  Time,  and  Number : 

Of  the  Ideas  on  which  the  Mechanical  Sciences  (as 
Mechanics,  Hydrostatics,  Physical  Astronomy)  more  pecu 
liarly  rest ;  the  ideas  of  Force  and  Matter •,  or  rather  the 
idea  of  Cause,  which  is  the  basis  of  these  : 

Of  the  Ideas  which  the  Secondary  Mechanical  Sciences 
(Acoustics,  Optics,  and  Thermotics)  involve ;  namely,  the 
Ideas  of  the  Externality  of  objects,  and  of  the  Media 
by  which  we  perceive  their  qualities : 

Of  the  Ideas  which  are  the  basis  of  Mechanico-chc- 
mical  and  Chemical  Science;  Polarity,  Chemical  Affinity, 
and  Substance ;  and  the  Idea  of  Symmetry,  a  necessary 
part  of  the  Philosophy  of  Crystallography : 

Of  the  Ideas  on  which  the  Classificatory  Sciences 
proceed  (Mineralogy,  Botany,  and  Zoology) ;  namely,  the 
Ideas  of  Resemblance,  and  of  its  gradations,  and  of 
Natural  Affinity: 

Finally,  of  those  Ideas  on  which  the  Physiological 
Sciences  are  founded ;  the  Ideas  of  separate  Vital  Powers, 
such  as  Assimilation  and  Irritability ;  and  the  Idea  of 
Final  Cause. 

We  have,  besides  these,  the  Palsetiological  Sciences, 
which  proceed  mainly  on  the  conception  of  Historical 

It  is  plain  that  when  we  have  proceeded  so  far  as 
this,  we  have  advanced  to  the  verge  of  those  speculations 
which  have  to  do  with  mind  as  well  as  body.  The 
extension  of  our  philosophy  to  such  a  field,  if  it  can  be 
justly  so  extended,  will  be  one  of  the  most  important 


results  of  our  researches;  but  on  that  very  account  we 
must  fully  study  the  lessons  which  we  learn  in  those 
fields  of  speculation  where  our  doctrines  are  most  secure, 
before  we  venture  into  a  region  where  our  principles  will 
appear  to  be  more  precarious,  and  where  they  are  inevi 
tably  less  precise. 

We  now  proceed  to  the  examination  of  the  above 
Ideas,  and  to  such  essays  towards  the  philosophy  of  each 
Science  as  this  course  of  investigation  may  suggest. 

VOL.  i.    w.  P. 






1.  ALL  external  objects  and  events  which  we  can  con 
template  are  viewed  as  having  relations  of  Space,  Time, 
and  Number ;  and  are  subject  to  the  general  conditions 
which  these  Ideas  impose,  as  well  as  to  the  particular 
laws  which  belong  to  each  class  of  objects  and  occur 
rences.  The  special  laws  of  nature,  considered  under 
the  various  aspects  which  constitute  the  different  sciences, 
are  obtained  by  a  mixed  reference  to  experience  and  to 
the  fundamental  ideas  of  each  science.  But  besides  the 
sciences  thus  formed  by  the  aid  of  special  experience,  the 
conditions  which  flow  from  those  more  comprehensive 
ideas  first  mentioned,  Space,  Time,  and  Number,  consti 
tute  a  body  of  science,  applicable  to  objects  and  changes 
of  all  kinds,  and  deduced  without  recurrence  being  had 
to  any  observation  in  particular.  These  sciences,  thus 
unfolded  out  of  ideas  alone,  unmixed  with  any  reference 
to  the  phenomena  of  matter,  are  hence  termed  Pure 
Sciences.  The  principal  sciences  of  this  class  are  Geome 
try,  Theoretical  Arithmetic,  and  Algebra  considered  in  its 
most  general  sense,  as  the  investigation  of  the  relations 
of  space  and  number  by  means  of  general  symbols. 


2.  These   Pure  Sciences  were  not  included  in  our 
survey  of  the  history  of  the  sciences,  because  they  are 
not  inductive  sciences.     Their  progress  has  riot  consisted 
in  collecting  laws  from  phenomena,  true  theories  from 
observed  facts,  and  more  general  from  more  limited  laws ; 
but  in  tracing  the  consequences  of  the  ideas  themselves, 
and  in  detecting  the  most  general  and  intimate  analogies 
and  connexions  which  prevail  among  such  conceptions  as 
are  derivable  from  the  ideas.     These  sciences  have  no 
principles  besides  definitions  and  axioms,  and  no  process 

I  of  proof  but  deduction ;  this  process,  however,  assuming 
here  a  most  remarkable  character ;  and  exhibiting  a  com 
bination  of  simplicity  and  complexity,  of  rigour  and 
generality,  quite  unparalleled  in  other  subjects. 

3.  The  universality  of  the  truths,  and  the  rigour  of 
the   demonstrations   of  these   pure   sciences,   attracted 
attention  in  the  earliest  times ;  and  it  was  perceived  that 
they  offered  an  exercise  and  a  discipline  of  the  intellec 
tual  faculties,  in  a  form  peculiarly  free  from  admixture 
of  extraneous  elements.     They  were  strenuously  culti 
vated  by  the  Greeks,  both  with  a  view  to  such  a  disci 
pline,  and  from  the  love  of  speculative  truth  which  pre 
vailed  among  that  people :  and  the  name  mathematics,  by 
which  they  are  designated,  indicates  this  their  character 
of  disciplinal  studies. 

4.  As  has  already  been  said,  the  ideas  which  these 
sciences  involve  extend  to  all  the  objects  and  changes 
which  we  observe  in  the  external  world ;  and  hence  the 
consideration  of  mathematical   relations   forms  a  large 
portion  of  many  of  the  sciences  which  treat  of  the  phe 
nomena   and   laws  of  external  nature,  as   Astronomy, 
Optics,  and  Mechanics.     Such  sciences  are  hence  often 
termed  Mixed  Mathematics,  the  relations  of  space  and 
number  being,  in  these  branches  of  knowledge,  combined 
with    principles    collected    from    special    observation ; 


while  Geometry,  Algebra,  and  the  like  subjects,  which 
involve  no  result  of  experience,  are  called  Pure  Mathe 

5.  Space,  time,  and  number,  may  be  conceived  as 
forms  by  which  the  knowledge  derived  from  our  sensa 
tions  is  moulded,  and  which  are  independent  of  the  dif 
ferences  in  the  matter  of  our  knowledge,  arising  from  the 
sensations  themselves.  Hence  the  sciences  which  have 
these  ideas  for  their  subject  may  be  termed  Formal 
Sciences.  In  this  point  of  view,  they  are  distinguished 
from  sciences  in  which,  besides  these  mere  formal  laws 
by  which  appearances  are  corrected,  we  endeavour  to 
apply  to  the  phenomena  the  idea  of  cause,  or  some  of  the 
other  ideas  which  penetrate  further  into  the  principles 
of  nature.  We  have  thus,  in  the  History,  distinguished 
Formal  Astronomy  and  Formal  Optics  from  Physical 
Astronomy  and  Physical  Optics. 

We  now  proceed  to  our  examination  of  the  Ideas 
which  constitute  the  foundation  of  these  formal  or  pure 
mathematical  sciences,  beginning  with  the  Idea  of  Space. 

OF    THE     IDEA    OF    SPACE. 

1.  BY  speaking  of  space  as  an  Idea,  I  intend  to  imply, 
as  has  already  been  stated,  that  the  apprehension  of 
objects  as  existing  in  space,  and  of  the  relations  of  posi 
tion,  &c.,  prevailing  among  them,  is  not  a  consequence 
of  experience,  but  a  result  of  a  peculiar  constitution  and 
activity  of  the  mind,  which  is  independent  of  all  expe 
rience  in  its  origin,  though  constantly  combined  with 
experience  in  its  exercise. 

That  the  idea  of  space  is  thus  independent  of  experi 
ence,  has  already  been  pointed  out  in  speaking  of  ideas 

OF  THE  IDEA   OF   SPACE.  85 

in  general :  but  it  may  be  useful  to  illustrate  the  doctrine 
further  in  this  particular  case. 

I  assert,  then,  that  space  is  not  a  notion  obtained 
by  experience.  Experience  gives  us  information  con 
cerning  things  without  us :  but  our  apprehending  them 
as  without  us,  takes  for  granted  their  existence  in  space. 
Experience  acquaints  us  what  are  the  form,  position, 
magnitude  of  particular  objects :  but  that  they  have  form, 
position,  magnitude,  presupposes  that  they  are  in  space. 
We  cannot  derive  from  appearances,  by  the  way  of 
observation,  the  habit  of  representing  things  to  ourselves 
as  in  space ;  for  no  single  act  of  observation  is  possible 
any  otherwise  than  by  beginning  with  such  a  representa 
tion,  and  conceiving  objects  as  already  existing  in  space. 

2.  That  our  mode  of  representing  space  to  ourselves 
is  not  derived  from  experience,  is  clear  also  from  this : 
— that  through  this  mode  of  representation  we  arrive  at 
propositions  which  are  rigorously  universal  and  neces 
sary.  Propositions  of  such  a  kind  could  not  possibly  be 
obtained  from  experience ;  for  experience  can  only  teach 
us  by  a  limited  number  of  examples,  and  therefore  can 
never  securely  establish  a  universal  proposition :  and 
again,  experience  can  only  inform  us  that  anything  is  so, 
and  can  never  prove  that  it  must  be  so.  That  two  sides 
of  a  triangle  are  greater  than  the  third  is  a  universal 
and  necessary  geometrical  truth:  it  is  true  of  all  tri 
angles  ;  it  is  true  in  such  a  way  that  the  contrary  cannot 
be  conceived.  Experience  could  not  prove  such  a  propo 
sition.  And  experience  has  not  proved  it ;  for  perhaps 
no  man  ever  made  the  trial  as  a  means  of  removing 
doubts :  and  no  trial  could,  in  fact,  add  in  the  smallest 
degree  to  the  certainty  of  this  truth.  To  seek  for  proof 
of  geometrical  propositions  by  an  appeal  to  observation 
proves  nothing  in  reality,  except  that  the  person  who 
has  recourse  to  such  grounds  has  no  due  apprehension 


of  the  nature  of  geometrical  demonstration.  We  have 
heard  of  persons  who  convinced  themselves  by  measure 
ment  that  the  geometrical  rule  respecting  the  squares 
on  the  sides  of  a  right-angled  triangle  was  true :  but 
these  were  persons  whose  minds  had  been  engrossed  by 
practical  habits,  and  in  whom  the  speculative  develope- 
ment  of  the  idea  of  space  had  been  stifled  by  other  em 
ployments.  The  practical  trial  of  the  rule  may  illustrate, 
but  cannot  prove  it.  The  rule  will  of  course  be  con 
firmed  by  such  trial,  because  what  is  true  in  general  is 
true  in  particular:  but  the  rule  cannot  be  proved  from  any 
number  of  trials,  for  no  accumulation  of  particular  cases 
makes  up  a  universal  case.  To  all  persons  who  can  see 
the  force  of  any  proof,  the  geometrical  rule  above  referred 
to  is  as  evident,  and  its  evidence  as  independent  of  ex 
perience,  as  the  assertion  that  sixteen  and  nine  make 
twenty-five.  At  the  same  time,  the  truth  of  the  geome 
trical  rule  is  quite  independent  of  numerical  truths,  and 
results  from  the  relations  of  space  alone.  This  could 
not  be  if  our  apprehension  of  the  relations  of  space  were 
the  fruit  of  experience :  for  experience  has  no  element 
from  which  such  truth  and  such  proof  could  arise. 

3.  Thus  the  existence  of  necessary  truths,  such  as 
those  of  geometry,  proves  that  the  idea  of  space  from 
which  they  flow,  is  not  derived  from  experience.  Such 
truths  are  inconceivable  on  the  supposition  of  their  being 
collected  from  observation ;  for  the  impressions  of  sense 
include  no  evidence  of  necessity.  But  we  can  readily 
understand  the  necessary  character  of  such  truths,  if  we 
conceive  that  there  are  certain  necessary  conditions  under 
which  alone  the  mind  receives  the  impressions  of  sense. 
Since  these  conditions  reside  in  the  constitution  of  the 
mind,  and  apply  to  every  perception  of  an  object  to 
which  the  mind  can  attain,  we  easily  see  that  their  rules 
must  include,  not  only  all  that  has  been,  but  all  that  can 

OF   THE    IDEA   OF   SPACE.  87 

be,  matter  of  experience.  Our  sensations  can  each  con 
vey  no  information  except  about  itself;  each  can  contain 
no  trace  of  another  additional  sensation ;  and  thus  no 
relation  and  connexion  between  two  sensations  can  be 
given  by  the  sensations  themselves.  But  the  mode  in 
which  the  mind  perceives  these  impressions  as  objects, 
may  and  will  introduce  necessary  relations  among  them : 
and  thus  by  conceiving  the  idea  of  space  to  be  a  con 
dition  of  perception  in  the  mind,  we  can  conceive  the 
existence  of  necessary  truths,  which  apply  to  all  per 
ceived  objects. 

4.  If  we  consider  the   impressions  of  sense  as  the 
mere  materials  of  our  experience,  such  materials  may 
be  accumulated  in  any  quantity  and  in  any  order.     But 
if  we  suppose  that  this  matter  has  a  certain  form  given 
it,  in  the  act  of  being  accepted  by  the  mind,   we  can 
understand  how  it  is  that  these  materials  are  subject  to 
inevitable  rules ; — how  nothing  can  be  perceived  exempt 
from  the  relations  which  belong  to  such  a  form.     And 
since  there  are  such  truths  applicable  to  our  experience, 
and  arising  from  the   nature   of  space,   we   may   thus 
consider  space  as  a  form  which  the  materials  given  by 
experience  necessarily  assume  in  the  mind;   as  an  ar 
rangement  derived  from  the  perceiving  mind,  and  not 
from  the  sensations  alone. 

5.  Thus  this  phrase, — that  space  is  &form  belonging 
to  our  perceptive  power, — may  be  employed  to  express 
that  we  cannot  perceive  objects  as  in  space,  without  an 
operation  of  the  mind  as  well  as  of  the  senses — without 
active  as  well  as   passive  faculties.     This  phrase,  how 
ever,  is  not  necessary  to  the  exposition  of  our  doctrines. 
Whether  we  call  the  conception  of  space  a  condition  of 
perception,  a  form  of  perception,  or  an  idea,  or  by  any 
other  term,  it  is  something  originally  inherent  in  the 
mind  perceiving,  and  not  in  the  objects  perceived.     And 


it  is  because  the  apprehension  of  all  objects  is  thus  sub 
jected  to  certain  mental  conditions,  forms  or  ideas,  that 
our  knowledge  involves  certain  inviolable  relations  and 
necessary  truths.  The  principles  of  such  truths,  so  far 
as  they  regard  space,  are  derived  from  the  idea  of  space> 
and  we  must  endeavour  to  exhibit  such  principles  in 
their  general  form.  But  before  we  do  this,  we  may 
notice  some  of  the  conditions  which  belong,  not  to  our 
Ideas  in  general,  but  to  this  Idea  of  Space  in  parti 




1.  SOME  of  the  Ideas  which  we  shall  have  to  examine 
involve  conceptions  of  certain  relations  of  objects,  as  the 
idea  of  Cause  and  of  Likeness ;  and  may  appear  to  be 
suggested  by  experience,  enabling  us  to  abstract  this 
general  relation  from  particular  cases.  But  it  will  be 
seen  that  Space  is  not  such  a  general  conception  of  a 
relation.  For  we  do  not  speak  of  Spaces  as  we  speak  of 
Causes  and  Likenesses,  but  of  Space.  And  when  wre 
speak  of  spaces,  we  understand  by  the  expression,  parts 
of  one  and  the  same  identical  every  where -extended 
Space.  We  conceive  a  Universal  Space ;  which  is  not 
made  up  of  these  partial  spaces  as  its  component  parts, 
for  it  would  remain  if  these  were  taken  away ;  and  these 
cannot  be  conceived  without  presupposing  absolute  space. 
Absolute  Space  is  essentially  one ;  and  the  complication 
which  exists  in  it,  and  the  conception  of  various  spaces, 
depends  merely  upon  boundaries.  Space  must,  there 
fore,  be,  as  we  have  said,  not  a  general  conception 
abstracted  from  particulars,  but  a  universal  mode  of 
representation,  altogether  independent  of  experience. 


2.  Space  is  infinite.    We  represent  it  to  ourselves  as 
an  infinitely  great  magnitude.     Such  an  idea  as  that  of 
Likeness  or  Cause,  is,  no  doubt,  found   in   an   infinite 
number  of  particular  cases,  and  so  far  includes  these 
cases.     But  these  ideas  do  riot  include  an  infinite  number 
of  cases  as  parts  of  an  infinite   whole.     When  we  say 
that  all  bodies  and  partial  spaces  exist  in  infinite  space, 
we  use  an  expression  which  is  not  applied  in  the  same 
sense  to  any  cases  except  those  of  Space  and  Time. 

3.  What  is  here  said  may  appear  to  be  a  denial  of 
the  real  existence  of  space.     It  must  be  observed,  how 
ever,  that  we  do  not   deny,  but   distinctly   assert,  the 
existence  of  space  as  a  real  and  necessary  condition  of 
all  objects  perceived ;  and  that  we  not  only  allow  that 
objects  are  seen  external  to  us,  but  we  found  upon  the 
fact  of  their  being  so  seen,  our  view  of  the  nature  of 
space.     If,  however,  it  be  said  that  we  deny  the  reality 
of  space  as  an  object  or  thing,  this  is  true.     Nor  does  it 
appear  easy  to  maintain  that  space  exists  as  a  thing, 
when  it  is  considered  that  this  thing  is  infinite  in  all  its 
dimensions;   and,  moreover,  that  it  is  a  thing,  which, 
being  nothing  in  itself,  exists  only  that  other  things  may 
exist  in  it.     And  those  who  maintain  the  real  existence 
of  space,  must  also  maintain  the  real  existence  of  time  in 
the  same  sense.     Now  two  infinite  things,  thus  really 
existing,  and  yet  existing  only  as  other  things  exist  in 
them,  are  notions  so  extravagant  that  we  are  driven  to 
some  other  mode  of  explaining  the  state  of  the  matter. 

4.  Thus  space  is  not  an  object  of  which  we  perceive 
the  properties,  but  a  form  of  our  perception;  not  a  thing 
which  aifects  our  senses,  but  an  idea  to  which  we  con 
form  the  impressions  of  sense.     And  its  peculiarities  ap 
pear  to  depend  upon  this,  that  it  is  not  only  a  form  of 
sensation,  but  of  intuition ;  that  in  reference  to  space, 
we  not  only  perceive  but  contemplate  objects.     We  see 


objects  in  space,  side  by  side,  exterior  to  each  other; 
space,  and  objects  in  so  far  as  they  occupy  space,  have 
parts  exterior  to  other  parts ;  and  have  the  whole  thus 
made  up  by  the  juxtaposition  of  parts.  This  mode  of 
apprehension  belongs  only  to  the  ideas  of  space  and 
time.  Space  and  Time  are  made  up  of  parts,  but  Cause 
and  Likeness  are  not  apprehended  as  made  up  of  parts. 
And  the  term  intuition  (in  its  rigorous  sense)  is  appli 
cable  only  to  that  mode  of  contemplation  in  which  we 
thus  look  at  objects  as  made  up  of  parts,  and  apprehend 
the  relations  of  those  parts  at  the  same  time  and  by  the 
same  act  by  which  we  apprehend  the  objects  themselves. 

5.  As  we  have  said,  space  limited  by  boundaries  gives 
rise  to  various  conceptions  which  we  have  often  to  con 
sider.     Thus  limited,  space  assumes/brw  or  figure;  and 
the  variety  of  conceptions  thus  brought  under  our  notice 
is  infinite.    We  have  every  possible  form  of  line,  straight 
line,  and  curve ;  and  of  curves  an  endless  number ; — cir 
cles,   parabolas,  hyperbolas,  spirals,  helices.     We  have 
plane  surfaces  of  various  shapes, — parallelograms,  poly 
gons,  ellipses ;  and  we  have  solid  figures, — cubes,  cones, 
cylinders,  spheres,  spheroids,  and  so  on.     All  these  have 
their  various  properties,  depending  on  the  relations  of 
their  boundaries ;  and  the  investigation  of  their  proper 
ties  forms  the  business  of  the  science  of  Geometry. 

6.  Space  has  three  dimensions,  or  directions  in  which 
it  may  be  measured ;  it  cannot  have  more  or  fewer.    The 
simplest  measurement  is  that  of  a  straight  line,  which 
has   length   alone.      A   surface    has   both   length    and 
breadth :  and  solid  space  has  length,  breadth,  and  thick 
ness  or  depth.     The  origin  of  such  a  difference  of  dimen 
sions  will  be  seen  if  we  reflect  that  each  portion  of  space 
lias  a  boundary,  and  is  extended  both  in  the  direction  in 
which  its  boundary  extends,  and  also  in  a  direction  from 
its  boundary ;  for  otherwise  it  would  not  be  a  boundary. 


A  point  has  no  dimensions.  A  line  has  but  one  dimen 
sion, — the  distance  from  its  boundary,  or  its  length.  A 
plane,  bounded  by  a  straight  line,  has  the  dimension 
which  belongs  to  this  line,  and  also  has  another  dimen 
sion  arising  from  the  distance  of  its  parts  from  this  bound 
ary  line ;  and  this  may  be  called  breadth.  A  solid, 
bounded  by  a  plane,  has  the  dimensions  which  this  plane 
has ;  and  has  also  a  third  dimension,  which  we  may  call 
height  or  depth,  as  we  consider  the  solid  extended  above 
or  below  the  plane ;  or  thickness,  if  we  omit  all  con 
sideration  of  up  and  down.  And  no  space  can  have  any 
dimensions  which  are  not  resoluble  into  these  three. 

We  may  now  proceed  to  consider  the  mode  in  which 
the  idea  of  space  is  employed  in  the  formation  of 



1.  THE  relations  of  space  have  been  apprehended 
with  peculiar  distinctness  and  clearness  from  the  very 
first  unfolding  of  man's  speculative  powers.  This  was  a 
consequence  of  the  circumstance  which  we  have  just 
noticed,  that  the  simplest  of  these  relations,  and  those  on 
which  the  others  depend,  are  seen  by  intuition.  Hence, 
as  soon  as  men  were  led  to  speculate  concerning  the 
relations  of  space,  they  assumed  just  principles,  and 
obtained  true  results.  It  is  said  that  the  science  of 
geometry  had  its  origin  in  Egypt,  before  the  dawn  of  the 
Greek  philosophy :  but  the  knowledge  of  the  early 
Egyptians  (exclusive  of  their  mythology)  appears  to  have 
been  purely  practical ;  and,  probably,  their  geometry 
consisted  only  in  some  maxims  of  la nd-measuring,  which 
is  what  the  term  implies.  The  Greeks  of  the  time  of 


Plato,  had,  however,  not  only  possessed  themselves  of 
many  of  the  most  remarkable  elementary  theorems  of 
the  science ;  but  had,  in  several  instances,  reached  the 
boundary  of  the  science  in  its  elementary  form  ;  as  when 
they  proposed  to  themselves  the  problems  of  doubling 
the  cube  and  squaring  the  circle. 

But  the  deduction  of  these  theorems  by  a  systematic 
process,  and  the  primary  exhibition  of  the  simplest  prin 
ciples  involved  in  the  idea  of  space,  which  such  a 
deduction  requires,  did  not  take  place,  so  far  as  we  are 
aware,  till  a  period  somewhat  later.  The  Elements  of 
Geometry  of  Euclid,  in  which  this  task  was  performed, 
are  to  this  day  the  standard  work  on  the  subject :  the 
author  of  this  work  taught  mathematics  with  great 
applause  at  Alexandria,  in  the  reign  of  Ptolemy  Lagus, 
about  280  years  before  Christ.  The  principles  which 
Euclid  makes  the  basis  of  his  system  have  been  very 
little  simplified  since  his  time ;  and  all  the  essays  and 
controversies  which  bear  upon  these  principles,  have 
had  a  reference  to  the  form  in  which  they  are  stated 
by  him. 

2.  Definitions. — The  first  principles  of  Euclid's  geo 
metry  are,  as  the  first  principles  of  any  system  of 
geometry  must  be,  definitions  and  axioms  respecting 
the  various  ideal  conceptions  which  he  introduces;  as 
straight  lines,  parallel  lines,  angles,  circles,  and  the  like. 
But  it  is  to  be  observed  that  these  definitions  and 
axioms  are  very  far  from  being  arbitrary  hypotheses  and 
assumptions.  They  have  their  origin  in  the  idea  of 
space,  and  are  merely  modes  of  exhibiting  that  idea  in 
such  a  manner  as  to  make  it  afford  grounds  of  deductive 
reasoning.  The  axioms  are  necessary  consequences  of 
the  conceptions  respecting  which  they  are  asserted ;  and 
the  definitions  are  no  less  necessary  limitations  of  con 
ceptions  ;  not  requisite  in  order  to  arrive  at  this  or  that 


consequence ;  but  necessary  in  order  that  it  may  be 
possible  to  draw  any  consequences,  and  to  establish  any 
general  truths. 

For  example,  if  we  rest  the  end  of  one  straight 
staff  upon  the  middle  of  another  straight  staff,  and  move 
the  first  staff  into  various  positions,  we,  by  so  doing, 
alter  the  angles  which  the  first  staff  makes  with  the 
other  to  the  right  hand  and  to  the  left.  But  if  we 
place  the  staff  in  that  special  position  in  which  these 
two  angles  are  equal,  each  of  them  is  a  right  angle, 
according  to  Euclid ;  and  this  is  the  definition  of  a  right 
angle,  except  that  Euclid  employs  the  abstract  con 
ception  of  straight  lines,  instead  of  speaking,  as  we  have 
done,  of  staves.  But  this  selection  of  the  case  in  which 
the  two  angles  are  equal  is  not  a  mere  act  of  caprice ; 
as  it  might  have  been  if  he  had  selected  a  case  in  which 
these  angles  are  unequal  in  any  proportion.  For  the 
consequences  which  can  be  drawn  concerning  the  cases 
of  unequal  angles,  do  not  lead  to  general  truths,  without 
some  reference  to  that  peculiar  case  in  which  the  angles 
are  equal :  and  thus  it  becomes  necessary  to  single  out 
and  define  that  special  case,  marking  it  by  a  special 
phrase.  And  this  definition  not  only  gives  complete  and 
distinct  knowledge  what  a  right  angle  is,  to  any  one 
who  can  form  the  conception  of  an  angle  in  general ;  but 
also  supplies  a  principle  from  which  all  the  properties  of 
right  angles  may,  be  deduced. 

3.  Axioms.— With  regard  to  other  conceptions  also, 
as  circles,  squares,  and  the  like,  it  is  possible  to  lay 
down  definitions  which  are  a  sufficient  basis  for  our 
reasoning,  so  far  as  such  figures  are  concerned.  But, 
besides  these  definitions,  it  has  been  found  necessary  to 
introduce  certain  axioms  among  the  fundamental  prin 
ciples  of  geometry.  These  are  of  the  simplest  character ; 
for  instance,  that  two  straight  lines  cannot  cut  each 


other  in  more  than  one  point,  and  an  axiom  concerning 
parallel  lines.  Like  the  definitions,  these  axioms  flow 
from  the  Idea  of  Space,  and  present  that  idea  under 
various  aspects.  They  are  different  from  the  definitions ; 
nor  can  the  definitions  be  made  to  take  the  place  of  the 
axioms  .in  the  reasoning  by  which  elementary  geo 
metrical  properties  are  established.  For  example,  the 
definition  of  parallel  straight  lines  is,  that  they  are  such 
as,  however  far  continued,  can  never  meet :  but,  in  order 
to  reason  concerning  such  lines,  we  must  further  adopt 
some  axiom  respecting  them :  for  example,  we  may  very 
conveniently  take  this  axiom ;  that  two  straight  lines 
which  cut  one  another  are  not  both  of  them  parallel  to 
a  third  straight  line"".  The  definition  and  the  axiom  are 
seen  to  be  inseparably  connected  by  our  intuition  of  the 
properties  of  space ;  but  the  axiom  cannot  be  proved 
from  the  definition,  by  any  rigorous  deductive  demon 
stration.  And  if  we  were  to  take  any  other  definition  of 
two  parallel  straight  lines,  (as  that  they  are  both  per 
pendicular  to  a  third  straight  line,)  we  should  still,  at 
some  point  or  other  of  our  progress,  fall  in  with  the 
same  difficulty  of  demonstratively  establishing  their  pro 
perties  without  some  further  assumption. 

4.  Thus  the  elementary  properties  of  figures,  which 
are  the  basis  of  our  geometry,  are  necessary  results  of 
our  Idea  of  Space ;  and  are  connected  with  each  other 
by  the  nature  of  that  idea,  and  not  merely  by  our  hypo 
theses  and  constructions.  Definitions  and  axioms  must 
be  combined,  in  order  to  express  this  idea  so  far  as 
the  purposes  of  demonstrative  reasoning  require.  These 
verbal  enunciations  of  the  results  of  the  idea  cannot  be 
made  to  depend  on  each  other  by  logical  consequence ; 
but  have  a  mutual  dependence  of  a  more  intimate  kind, 

*  This  axiom  is  simpler  and  more  convenient  than  that  of  Euclid. 
It  is  employed  by  the  late  Professor  Flayfair  in  his  Geometry. 


which  words  cannot  fully  convey.  It  is  not  possible  to 
resolve  these  truths  into  certain  hypotheses,  of  which  all 
the  rest  shall  be  the  necessary  logical  consequence.  The 
necessity  is  not  hypothetical,  but  intuitive.  The  axioms 
require  not  to  be  granted,  but  to  be  seen.  If  any  one 
were  to  assent  to  them  without  seeing  them  to  be  true, 
his  assent  would  be  of  no  avail  for  purposes  of  reason 
ing:  for  he  would  be  also  unable  to  see  in  what  cases 
they  might  be  applied.  The  clear  possession  of  the 
Idea  of  Space  is  the  first  requisite  for  all  geometrical 
reasoning ;  and  this  clearness  of  idea  may  be  tested  by 
examining  whether  the  axioms  offer  themselves  to  the 
mind  as  evident. 

5.  The  necessity  of  ideas  added  to  sensations,  in 
order  to  produce  knowledge,  has  often  been  overlooked 
or  denied  in  modern  times.  The  ground  of  necessary 
truth  which  ideas  supply  being  thus  lost,  it  was  con 
ceived  that  there  still  remained  a  ground  of  necessity  in. 
definitions; — that  we  might  have  necessary  truths,  by 
asserting  especially  what  the  definition  implicity  involved 
in  general.  It  was  held,  also,  that  this  was  the  case  in 
geometry: — that  all  the  properties  of  a  circle,  for 
instance,  were  implicitly  contained  in  the  definition  of  a 
circle.  That  this  alone  is  not  the  ground  of  the  neces 
sity  of  the  truths  which  regard  the  circle, — that  we 
could  not  in  this  way  unfold  a  definition  into  propor 
tions,  without  possessing  an  intuition  of  the  relations  to 
which  the  definition  led, — has  already  been  shown.  But 
the  insufficiency  of  the  above  account  of  the  grounds  of 
necessary  geometrical  truth  appeared  in  another  way 
also.  It  was  found  impossible  to  lay  down  a  system  of 
definitions  out  of  which  alone  the  whole  of  geometrical 
truth  could  be  evolved.  It  was  found  that  axioms  could 
not  be  superseded.  No  definition  of  a  straight  line 
could  be  given  which  rendered  the  axiom  concerning 


straight  lines  superfluous.  And  thus  it  appeared  that 
the  source  of  geometrical  truths  was  not  definition 
alone ;  and  we  find  in  this  result  a  confirmation  of  the 
doctrine  which  we  are  here  urging,  that  this  source  of 
truth  is  to  be  found  in  the  form  or  conditions  of  our 
perception ; — in  the  idea  which  we  unavoidably  combine 
with  the  impressions  of  sense ; — in  the  activity,  and  not 
in  the  passivity  of  the  mind"-. 

6.  This  will  appear  further  when  we  come  to  con 
sider  the  mode  in  which  \ve  exercise  our  observation 
upon  the  relations  of  space.  But  we  may,  in  the  first 
place,  make  a  remark  which  tends  to  show  the  con 
nexion  between  our  conception  of  a  straight  line,  and 
the  axiom  which  is  made  the  foundation  of  our  reason 
ings  concerning  space.  The  axiom  is  this; — that  two 
straight  lines,  which  have  both  their  ends  joined,  cannot 
have  the  intervening  parts  separated  so  as  to  inclose  a 
space.  The  necessity  of  this  axiom  is  of  exactly  the 
same  kind  as  the  necessity  of  the  definition  of  a  right 
angle,  of  which  we  have  already  spoken.  For  as  the  line 
standing  on  another  makes  right  angles  when  it  makes 
the  angles  on  the  two  sides  of  it  equal;  so  a  line  is  a 
straight  line  when  it  makes  the  two  portions  of  space, 
on  the  two  sides  of  it,  similar.  And  as  there  is  only  a 
single  position  of  the  line  first  mentioned,  which  can 
make  the  angles  equal,  so  there  is  only  a  single  form  of 
a  line  which  can  make  the  spaces  near  the  line  similar 
on  one  side  and  on  the  other :  and  therefore  there  can 
not  be  two  straight  lines,  such  as  the  axiom  describes, 

*  I  formerly  stated  views  similar  to  these  in  some  "  Remarks" 
appended  to  a  work  which  I  termed  The  Mechanical  Euclid,  pub 
lished  in  1837-  These  Remarks,  so  far  as  they  bear  upon  the  question 
here  discussed,  were  noticed  and  controverted  in  No.  135  of  the  Edin 
burgh  Review.  As  an  examination  of  the  reviewer's  objections  may 
serve  further  to  illustrate  the  subject,  I  shall  annex  to  this  chapter  an 
answer  to  the  article  to  which  I  have  referred. 


which,  between  the  same  limits,  give  t\vo  different 
boundaries  to  space  thus  separated.  And  thus  we  see  a 
reason  for  the  axiom.  Perhaps  this  view  may  be  further 
elucidated  if  we  take  a  leaf  of  paper,  double  it,  and 
crease  the  folded  edge.  We  shall  thus  obtain  a  straight 
line  at  the  folded  edge ;  and  this  line  divides  the  surface 
of  the  paper,  as  it  was  originally  spread  out,  into  two 
similar  spaces.  And  that  these  spaces  are  similar  so  far 
as  the  fold  which  separates  them  is  concerned,  appears 
from  this ; — that  these  two  parts  coincide  when  the 
paper  is  doubled.  And  thus  a  fold  in  a  sheet  of  paper 
at  the  same  time  illustrates  the  definition  of  a  straight 
line  according  to  the  above  view,  and  confirms  the 
axiom  that  two  such  lines  cannot  enclose  a  space. 

If  the  separation  of  the  two  parts  of  space  were  made 
by  any  other  than  a  straight  line;  if,  for  instance,  the 
paper  were  cut  by  a  concave  line ;  then,  on  turning  one 
of  the  parts  over,  it  is  easy  to  see  that  the  edge  of  one 
part  being  concave  one  way,  and  the  edge  of  the  other 
part  concave  the  other  way,  these  two  lines  would 
enclose  a  space.  And  each  of  them  would  divide  the 
whole  space  into  two  portions  which  were  not  similar ; 
for  one  portion  would  have  a  concave  edge,  and  the 
other  a  convex  edge.  Between  any  two  points,  there 
might  be  innumerable  lines  drawn,  some,  convex  one 
way,  and  some,  convex  the  other  way ;  but  the  straight 
line  is  the  line  which  is  not  convex  either  one  way  or 
the  other ;  it  is  the  single  medium  standard  from  which 
the  others  may  deviate  in  opposite  directions. 

Such  considerations  as  these  show  sufficiently  that 
the  singleness  of  the  straight  line  which  connects  any 
two  points  is  a  result  of  our  fundamental  conceptions  of 
space.  But  yet  the  above  conceptions  of  the  similar 
form  of  the  two  parts  of  space  on  the  two  sides  of  a  line, 
and  of  the  form  of  a  line  which  is  intermediate  among 

VOL.  i.    w.  p.  H 


all  other  forms,  are  of  so  vague  a  nature,  that  they  can 
not  fitly  be  made  the  basis  of  our  elementary  geometry ; 
and  they  are  far  more  conveniently  replaced,  as  they 
have  been  in  almost  all  treatises  of  geometry,  by  the 
axiom,  that  two  straight  lines  cannot  inclose  a  space. 

7.  But  we  may  remark  that,  in  what  precedes,  we 
have  considered  space  only  under  one  of  its  aspects : — as 
a  plane.  The  sheet  of  paper  wrhich  we  assumed  in  order 
to  illustrate  the  nature  of  a  straight  line,  was  supposed 
to  be  perfectly  plane  m  flat :  for  otherwise,  by  folding  it, 
we  might  obtain  a  line  not  straight.  TSow  this  assump 
tion  of  a  plane  appears  to  take  for  granted  that  very 
conception  of  a  straight  line  which  the  sheet  was  em 
ployed  to  illustrate ;  for  the  definition  of  a  plane  given 
in  the  Elements  of  Geometry  is,  that  it  is  a  surface  on 
which  lie  all  straight  lines  drawn  from  one  point  of  the 
surface  to  another.  And  thus  the  explanation  above 
given  of  the  nature  of  a  straight  line, — that  it  divides  a 
plane  space  into  similar  portions  on  each  side, — appears 
to  be  imperfect  or  nugatory. 

To  this  we  reply,  that  the  explanation  must  be  ren 
dered  complete  and  valid  by  deriving  the  conception  of 
a  plane  from  considerations  of  the  same  kind  as  those 
which  we  employed  for  a  straight  line.  Any  portion  of 
solid  space  may  be  divided  into  two  portions  by  surfaces 
passing  through  any  given  line  or  boundaries.  And 
these  surfaces  may  be  convex  either  on  one  side  or  on 
the  other,  and  they  admit  of  innumerable  changes  from 
being  convex  on  one  side  to  being  convex  on  the  other 
in  any  degree.  So  long  as  the  surface  is  convex  either 
way,  the  two  portions  of  space  which  it  separates  are  not 
similar,  one  having  a  convex  and  the  other  a  concave 
boundary.  But  there  is  a  certain  intermediate  position  of 
the  surface,  in  which  position  the  two  portions  of  space 
which  it  divides  have  their  boundaries  exactly  similar. 


Iii  this  position,  the  surface  is  neither  convex  nor  concave, 
but  plane.  And  thus  a  plane  surface  is  determined  by 
this  condition — of  its  being  that  single  surface  which  is 
the  intermediate  form  among  all  convex  and  concave 
surfaces  by  which  solid  space  can  be  divided, — and  of 
its  separating  such  space  into  two  portions,  of  which 
the  boundaries,  though  they  are  the  same  surface  in 
two  opposite  positions,  are  exactly  similar. 

Thus  a  plane  is  the  simplest  and  most  symmetrical 
boundary  by  which  a  solid  can  be  divided ;  and  a  straight 
line  is  the  simplest  and  most  symmetrical  boundary  by 
which  a  plane  can  be  separated.  These  conceptions  are 
obtained  by  considering  the  boundaries  of  an  intermin 
able  space,  capable  of  imaginary  division  in  every  direc 
tion.  And  as  a  limited  space  may  be  separated  into  two 
parts  by  a  plane,  and  a  plane  again  separated  into  two 
parts  by  a  straight  line,  so  a  line  is  divided  into  two  por 
tions  by  a  point,  which  is  the  common  boundary  of  the 
t\vo  portions ;  the  end  of  the  one  and  the  beginning  of  the 
other  portion  having  itself  no  magnitude,  form,  or  parts. 

8.  The  geometrical  properties  of  planes  and  solids 
are  deducible  from  the  first  principles  of  the  Elements, 
without  any  new  axioms ;  the  definition  of  a  plane  above 
quoted, — that  all  straight  lines  joining  its  points  lie  in 
the  plane, — being  a  sufficient  basis  for  all  reasoning  upon 
these  subjects.  And  thus,  the  views  which  we  have  pre 
sented  of  the  nature  of  space  being  verbally  expressed 
by  means  of  certain  definitions  and  axioms,  become  the 
groundwork  of  a  long  series  of  deductive  reasoning,  by 
which  is  established  a  very  large  and  curious  collection 
of  truths,  namely,  the  whole  science  of  Elementary 
Plane  and  Solid  Geometry. 

This  science  is  one  of  indispensable  use  and  constant 
reference,  for  every  student  of  the  laws  of  nature ;  for  the 
relations  of  space  and  number  are  the  alphabet  in  which 

II  2 


those  laws  are  written.  But  besides  the  interest  and  im 
portance  of  this  kind  which  geometry  possesses,  it  has  a 
great  and  peculiar  value  for  all  who  wish  to  understand 
the  foundations  of  human  knowledge,  and  the  methods 
by  which  it  is  acquired.  For  the  student  of  geometry 
acquires,  with  a  degree  of  insight  and  clearness  which 
the  unmathematical  reader  can  but  feebly  imagine,  a 
conviction  that  there  are  necessary  truths,  many  of  them 
of  a  very  complex  and  striking  character ;  and  that  a 
few  of  the  most  simple  and  self-evident  truths  which  it  is 
possible  for  the  mind  of  man  to  apprehend,  may,  by 
systematic  deduction,  lead  to  the  most  remote  and  unex 
pected  results. 

In  pursuing  such  philosophical  researches  as  that 
in  which  we  are  now  engaged,  it  is  of  great  advantage 
to  the  speculator  to  have  cultivated  to  some  extent  the 
study  of  geometry ;  since  by  this  study  he  may  become 
fully  aware  of  such  features  in  human  knowledge  as 
those  which  we  have  mentioned.  By  the  aid  of  the 
lesson  thus  learned  from  the  contemplation  of  geome 
trical  truths,  wre  have  been  endeavouring  to  establish 
those  further  doctrines; — that  these  truths  are  but  dif 
ferent  aspects  of  the  same  Fundamental  Idea,  and  that 
the  grounds  of  the  necessity  which  these  truths  possess 
reside  in  the  Idea  from  which  they  flow,  this  Idea  not 
being  a  derivative  result  of  experience,  but  its  primary 
rule.  When  the  reader  has  obtained  a  clear  and  satis 
factory  view  of  these  doctrines,  so  far  as  they  are  appli 
cable  to  our  knowledge  concerning  space,  he  has,  we  may 
trust,  overcome  the  main  difficulty  which  will  occur  in 
following  the  course  of  the  speculations  now  presented 
to  him.  He  is  then  prepared  to  go  forwards  with  us ;  to 
see  over  how  wide  a  field  the  same  doctrines  are  appli 
cable:  and  how  rich  and  various  a  harvest  of  knowledge 
springs  from  these  seemingly  scanty  principles. 


But  before  we  quit  the  subject  now  under  our  con 
sideration,  we  shall  endeavour  to  answer  some  objections 
which  have  been  made  to  the  views  here  presented;  and 
shall  attempt  to  illustrate  further  the  active  powers  which 
we  have  ascribed  to  the  mind. 



THE  Edinburgh  Review,  No.  cxxxv.,  contains  a  cri 
tique  on  a  work  termed  The  Mechanical  Euclid,  in  which 
opinions  were  delivered  to  nearly  the  same  effect  as  some 
of  those  stated  in  the  last  chapter,  and  in  Chapter  xi. 
of  the  First  Book.  Although  I  believe  that  there  are  no 
arguments  used  by  the  reviewer  to  which  the  answers 
will  not  suggest  themselves  in  the  mind  of  any  one  who 
has  read  with  attention  what  has  been  said  in  the  pre 
ceding  chapters  (except,  perhaps,  one  or  two  remarks 
which  have  reference  to  mechanical  ideas),  it  may  serve  to 

*  In  order  to  render  the  present  chapter  more  intelligible,  it  may 
be  proper  to  state  briefly  the  arguments  which  gave  occasion  to  the 
review.  After  noticing  Stewart's  assertions,  that  the  certainty  of  mathe 
matical  reasoning  arises  from  its  depending  upon  definitions,  and  that 
mathematical  truth  is  hypothetical;  I  urged, — that  no  one  has  yet 
been  able  to  construct  a  system  of  mathematical  truths  by  the  aid  of 
definitions  alone ;  that  a  definition  would  not  be  admissible  or  appli 
cable  except  it  agreed  with  a  distinct  conception  in  the  mind ;  that  the 
definitions  which  we  employ  in  mathematics  are  not  arbitrary  or  hypo 
thetical,  but  necessary  definitions;  that  if  Stewart  had  taken  as  his 
examples  of  axioms  the  peculiar  geometrical  axioms,  his  assertions 
would  have  been  obviously  erroneous ;  and  that  the  real  foundation  of 
the  truths  of  mathematics  is  the  Idea  of  Space,  which  may  be  expressed 
(for  purposes  of  demonstration)  partly  by  definitions  and  partly  by 


illustrate  the  subject  if  I  reply  to  the  objections  directly, 
taking  them  as  the  reviewer  has  stated  them. 

1.  I  had  dissented  from  Stewart's  assertion  that 
mathematical  truth  is  hypothetical,  or  depends  upon  arbi 
trary  definitions ;  since  we  understand  by  an  hypothesis 
a  supposition,  not  only  which  we  may  make,  but  may 
abstain  from  making,  or  may  replace  by  a  different  sup 
position  ;  whereas  the  definitions  and  hypotheses  of  geo 
metry  are  necessarily  such  as  they  are,  and  cannot  be 
altered  or  excluded.  The  reviewer  (p.  84),  informs  us 
that  he  understands  Stewart,  when  he  speaks  of  hypo 
theses  and  definitions  being  the  foundation  of  geometry, 
to  speak  of  the  hypothesis  that  real  objects  correspond 
to  our  geometrical  definitions.  "  If  a  crystal  be  an  exact 
hexahedron,  the  geometrical  properties  of  the  hexahe 
dron  may  be  predicated  of  that  crystal."  To  this  I  reply, 
— that  such  hypotheses  as  this  are  the  grounds  of  our 
applications  of  geometrical  truths  to  real  objects,  but 
can  in  no  way  be  said  to  be  the  foundation  of  the  truths 
themselves; — that  I  do  not  think  that  the  sense  which  the 
reviewer  gives  was  Stewart's  meaning; — but  that  if  it  was, 
this  view  of  the  use  of  mathematics  does  not  at  all  affect 
the  question  which  both  he  and  I  proposed  to  discuss, 
which  was,  the  ground  of  mathematical  certainty.  I  may 
add,  that  whether  a  crystal  be  an  exact  hexahedron,  is 
a  matter  of  observation  and  measurement,  not  of  defini 
tion.  I  think  the  reader  can  have  no  difficulty  in  seeing 
how  little  my  doctrine  is  affected  by  the  connexion  on 
which  the  reviewer  thus  insists.  I  have  asserted  that  the 
proposition  which  affirms  the  square  on  the  diagonal  of 
a  rectangle  to  be  equal  to  the  squares  on  two  sides,  does- 
not  rest  upon  arbitrary  hypotheses;  the  objector  answers, 
that  the  proposition  that  the  square  on  the  diagonal  of 
this  page  is  equal  to  the  squares  on  the  sides,  depends 
upon  the  arbitrary  hypothesis  that  the  page  is  a  rect- 


angle.  Even  if  this  fact  were  a  matter  of  arbitrary 
hypothesis,  what  could  it  have  to  do  with  the  general 
geometrical  proposition?  How  could  a  single  fact,  ob 
served  or  hypothetical,  affect  a  universal  and  necessary 
truth,  which  would  be  equally  true  if  the  fact  were  false? 
If  there  be  nothing  arbitrary  or  hypothetical  in  geometry 
till  we  come  to  such  steps  in  its  application,  it  is  plain 
that  the  truths  themselves  are  not  hypothetical;  which  is 
the  question  for  us  to  decide. 

2.  The  reviewer  then  (p.  85),  considers  the  doctrine 
that  axioms  as  well  as  definitions  are  the  foundations  of 
geometry;  and  here  he  strangely  narrows  and  confuses 
the  discussion  by  making  himself  the  advocate  of  Stewart, 
instead  of  arguing  the  question  itself.  I  had  asserted 
that  some  axioms  are  necessary  as  the  foundations  of 
mathematical  reasoning,  in  addition  to  the  definitions. 
If  Stewart  did  not  intend  to  discuss  this  question,  I  had 
no  concern  with  what  he  had  said  about  axioms.  But  I 
had  every  reason  to  believe  that  this  was  the  question 
which  Stewart  did  intend  to  discuss.  I  conceive  there  is 
no  doubt  that  he  intended  to  give  an  opinion  upon  the 
grounds  of  mathematical  reasoning  in  general.  For  he 
begins  his  discussions  (Elements,  Vol.  n.,  p.  38)  by  contest 
ing  Reid's  opinion  on  this  subject,  which  is  stated  gene 
rally;  and  he  refers  again  to  the  same  subject,  asserting 
in  general  terms,  that  the  first  principles  of  mathematics 
are  not  axioms  but  definitions.  If,  then,  afterwards,  he 
made  his  proof  narrower  than  his  assertion ; — if  having 
declared  that  no  axioms  are  necessary,  he  afterwards 
limited  himself  to  showing  that  seven  out  of  twelve  of 
Euclid's  axioms  arc  barren  truisms,  it  was  no  concern  of 
mine  to  contest  this  assertion,  which  left  my  thesis  un 
touched.  I  had  asserted  that  the  proper  geometrical 
axioms  (that  two  straight  lines  cannot  inclose  a  space, 
and  the  axiom  about  parallel  lines)  are  indispensable  in 


geometry.  What  account  the  reviewer  gives  of  these 
axioms  we  shall  soon  see;  but  if  Stewart  allowed  them  to 
be  axioms  necessary  to  geometrical  reasoning,  he  over 
turned  his  own  assertion  as  to  the  foundations  of  such 
reasoning ;  and  if  he  said  nothing  decisive  about  these 
axioms,  which  are  the  points  on  which  the  battle  must 
turn,  he  left  his  assertion  altogether  improved ;  nor  was 
it  necessary  for  me  to  pursue  the  war  into  a  barren  and 
unimportant  corner,  when  the  metropolis  was  surrendered. 
The  reviewer's  exultation  that  I  have  not  contested  the 
first  seven  axioms  is  an  amusing  example  of  the  self- 
complacent  zeal  of  advocacy. 

3.  But  let  us  turn  to  the  material  point, — the  proper 
geometrical  axioms.  What  is  the  reviewer's  account  of 
these?  Which  side  of  the  alternative  does  he  adopt? 
Do  they  depend  upon  the  definitions,  and  is  he  prepared 
to  show  the  dependence  ?  Or  are  they  superfluous,  and 
can  he  erect  the  structure  of  geometry  without  their  aid? 
One  of  these  two  courses,  it  would  seem,  he  must  take. 
For  we  both  begin  by  asserting  the  excellence  of  geo 
metry  as  an  example  of  demonstrated  truth.  It  is 
precisely  this  attribute  which  gives  an  interest  to  our 
present  inquiry.  How,  then,  does  the  reviewer  explain 
this  excellence  on  his  views  ?  How  does  he  reckon  the 
foundation  courses  of  the  edifice  which  we  agree  in  con 
sidering  as  a  perfect  example  of  intellectual  building  ? 

I  presume  I  may  take,  as  his  answer  to  this  question, 
his  hypothetical  statement  of  what  Stewart  would  have 
said,  (p.  87,)  on  the  supposition  that  there  had  been, 
among  the  foundations  of  geometry,  self-evident  indemon 
strable  truths :  although  it  is  certainly  strange  that  the 
reviewer  should  not  venture  to  make  up  his  mind  as  to 
the  truth  or  falsehood  of  this  supposition.  If  there  were 
such  truths  they  would  be,  he  says,  "  legitimate  filiations" 
of  the  definitions.  They  would  be  involved  in  the  defi- 


nitions.  And  again  he  speaks  of  the  foundation  of  the 
geometrical  doctrine  of  parallels  as  a  flaw,  and  as  a 
truth  which  requires,  but  has  not  received  demonstration. 
And  yet  again,  he  tells  us  that  each  of  these  supposed 
axioms  (Euclid's  twelfth,  for  instance),  is  "merely  an 
indication  of  the  point  at  which  geometry  fails  to  per 
form  that  which  it  undertakes  to  perform"  (p.  91);  and 
that  in  reality  her  truths  are  not  yet  demonstrated.  The 
amount  of  this  is,  that  the  geometrical  axioms  are  to  be 
held  to  be  legitimate  filiations  of  the  definitions,  because 
though  certainly  true,  they  cannot  be  proved  from  the 
definitions ;  that  they  are  involved  in  the  definitions, 
although  they  cannot  be  evolved  out  of  them ;  and  that 
rather  than  admit  that  they  have  any  other  origin  than 
the  definitions,  we  are  to  proclaim  that  geometry  has 
failed  to  perform  what  she  undertakes  to  perform. 

-\fo  this  I  reply — that  I  cannot  understand  what  is 
meant  by  "legitimate  filiations"  of  principles,  if  the  phrase 
not  mean  consequences  of  such  principles  established  by 
rigorous  and  formal  demonstrations ; — that  the  reviewer, 
if  he  claims  any  real  signification  for  his  phrase,  must 
substantiate  the  meaning  of  it  by  such  a  demonstration ; 
he  must  establish  his  "  legitimate  filiation"  by  a  genea 
logical  table  in  a  satisfactory  form.  "When  this  cannot 
be  done,  to  assert,  notwithstanding,  that  the  propositions 
are  involved  in  the  definitions,  is  a  mere  begging  the 
question;  and  to  excuse  this  defect  by  saying  that  geo 
metry  fails  to  perform  what  she  has  promised,  is  to  calum 
niate  the  character  of  that  science  which  we  profess  to 
make  our  standard,  rather  than  abandon  an  arbitrary 
and  unproved  assertion  respecting  the  real  grounds  of 
her  excellence.  I  add,  further,  that  if  the  doctrine  of 
parallel  lines,  or  any  other  geometrical  doctrine  of  which 
we  see  the  truth,  with  the  most  perfect  insight  of  its 
necessity,  have  not  hitherto  received  demonstration  to  the 


satisfaction  of  any  school  of  reasoners,  the  defect  must 
arise  from  their  erroneous  views  of  the  nature  of  demon 
strations,  and  the  grounds  of  mathematical  certainty. 

4.  I  conceive,  then,  that  the  reviewer  has  failed  alto 
gether  to  disprove  the  doctrine  that  the  axioms  of  geo 
metry  are  necessary  as  a  part  of  the  foundations  of  the 
science.    I  had  asserted  further  that  these  axioms  supply 
what  the  definitions  leave  deficient ;  and  that  they,  along 
with  definitions,  serve  to  present  the  idea  of  space  under 
such  aspects  that  we  can  reason  logically  concerning  it. 
To  this  the  reviewer  opposes  (p.  96)  the  common  opinion 
that  a  perfect  definition  is  a  complete  explanation  of  a 
name,   and  that  the  test  of  its  perfection  is,  that  we 
may   substitute  the    definition  for  the  name  wherever 
it  occurs.     I  reply,  that  my  doctrine,  that  a  definition 
expresses  a  part,  but  not  the  whole,  of  the  essential  cha 
racters  of  an  idea,  is  certainly  at  variance  with  an  opinion 
sometimes  maintained,  that  a  definition  merely  explains 
a  word,  and  should  explain  it  so  fully  that  it  may  always 
replace  it.  The  error  of  this  common  opinion  may,  I  think, 
be  shown  from  considerations  such  as  these ; — that  if  wre 
undertake  to  explain  one  word  by  several,  we  may  be 
called  upon,  on  the  same  ground,  to  explain  each  of  these 
several  by  others,  and  that  in  this  way  we  can  reach  no 
limit  nor  resting-place  ; — that  in  point  of  fact,  it  is  not 
found  to  lead  to  clearness,  but  to  obscurity,  when  in  the 
discussion  of  general  principles,  we  thus  substitute  defi 
nitions  for  single  terms ; — that  even  if  this  be  done,  we 
cannot  reason  without  conceiving  what  the  terms  mean ; 

—and  that,  in  doing  this,  the  relations  of  our  concep 
tions,  and  not  the  arbitrary  equivalence  of  two  forms  of 
expression,  are  the  foundations  of  our  reasoning. 

5.  The  reviewer  conceives  that  some  of  the  so-called 
axioms  are  really  definitions.     The  axiom,  that  "  magni 
tudes  which  coincide  with  each  other,  that  is,  which  fill 


the  same  space,  are  equal,"  is  a  definition  of  geometrical 
equality :  the  axiom,  that  "  the  whole  is  greater  than  its 
part,"  is  a  definition  of  whole  and  part.  But  surely  there 
are  very  serious  objections  to  this  view.  It  would  seem 
more  natural  to  say,  if  the  former  axiom  is  a  definition 
of  the  word  equal,  that  the  latter  is  a  definition  of  the 
word  greater.  And  how  can  one  short  phrase  define  two 
terms  ?  If  I  say,  "  the  heat  of  summer  is  greater  than 
the  heat  of  winter,"  does  this  assertion  define  anything, 
though  the  proposition  is  perfectly  intelligible  and  dis 
tinct?  I  think,  then,  that  this  attempt  to  reduce  these 
axioms  to  definitions  is  quite  untenable. 

6.  I  have  stated  that  a  definition  can  be  of  no  use, 
except  we  can  conceive  the  possibility  and  truth  of  the 
property  connected  with  it ;  and  that  if  we  do  conceive 
this,  we  may  rightly  begin  our  reasonings  by  stating  the 
property  as  an  axiom ;  which  Euclid  does,  in  the  case  of 
straight  lines  and  of  parallels.  The  reviewer  inquires, 
(p.  92,)  whether  I  am  prepared  to  extend  this  doctrine  to 
the  case  of  circles,  for  which  the  reasoning  is  usually 
rested  upon  the  definition ; — whether  I  would  replace  this 
definition  by  an  axiom,  asserting  the  possibility  of  such  a 
circle.  To  this  I  might  reply,  that  it  is  not  at  all  incum 
bent  upon  me  to  assent  to  such  a  change ;  for  I  have  all 
along  stated  that  it  is  indifferent  whether  the  fundamen 
tal  properties  from  which  we  reason  be  exhibited  as  defi 
nitions  or  as  axioms,  provided  their  necessity  be  clearly 
seen.  But  I  am  ready  to  declare  that  I  think  the  form 
of  our  geometry  would  be  not  at  all  the  worse,  if,  instead 
of  the  usual  definition  of  a  circle, — "  that  it  is  a  figure 
contained  by  one  line,  which  is  called  the  circumference, 
and  which  is  such,  that  all  straight  lines  drawn  from  a 
certain  point  within  the  circumference  are  equal  to  one 
another," — we  were  to  substitute  an  axiom  and  a  defini 
tion,  as  follows : — 


Axiom.  If  a  line  be  drawn  so  as  to  be  at  every  point 
equally  distant  from  a  certain  point,  this  line  will  return 
into  itself,  or  will  be  one  line  including  a  space. 

Definition.  The  space  is  called  a  circle,  the  line  the 
circumference,  and  the  point  the  center. 

And  this  being  done,  it  would  be  true,  as  the  reviewer 
remarks,  that  geometry  cannot  stir  one  step  without 
resting  on  an  axiom.  And  I  do  not  at  all  hesitate  to  say, 
that  the  above  axiom,  expressed  or  understood,  is  no  less 
necessary  than  the  definition,  and  is  tacitly  assumed  in 
every  proposition  into  which  circles  enter. 

7.  I  have,  I  think,  now  disposed  of  the  principal 
objections  which  bear  upon  the  proper  axioms  of  geo 
metry.  The  principles  which  are  stated  as  the  first  seven 
axioms  of  Euclid's  Elements,  need  not,  as  I  have  said,  be 
here  discussed.  They  are  principles  which  refer,  not  to 
Space  in  particular,  but  to  Quantity  in  general :  such, 
for  instance,  as  these ;  "  If  equals  be  added  to  equals  the 
wholes  are  equal ;" — "  If  equals  be  taken  from  equals 
the  remainders  are  equal."  But  I  will  make  an  obser 
vation  or  two  upon  them  before  I  proceed. 

Both  Locke  and  Stewart  have  spoken  of  these  axioms 
as  barren  truisms :  as  propositions  from  which  it  is  not 
possible  to  deduce  a  single  inference :  and  the  reviewer 
asserts  that  they  are  not  first  principles,  but  laws  of 
thought,  (p.  88.)  To  this  last  expression  I  am  willing 
to  assent ;  but  I  would  add,  that  not  only  these,  but  all 
the  principles  which  express  the  fundamental  conditions 
of  our  knowledge,  may  with  equal  propriety  be  termed 
laws  of  thought ;  for  these  principles  depend  upon  our 
ideas,  and  regulate  the  active  operations  of  the  mind,  by 
which  coherence  and  connexion  are  given  to  its  passive 
impressions.  But  the  assertion  that  no  conclusions  can 
be  drawn  from  simple  axioms,  or  laws  of  human  thought, 
which  regard  quantity,  is  by  no  means  true.  The  whole 


of  arithmetic, — for  instance,  the  rules  for  the  multiplica-  \ 
tion  and  division  of  large  numbers,  for  finding  a  common 
measure,  and,  in  short,  a  vast  body  of  theory  respecting 
numbers, — rests  upon  no  other  foundation  than  such 
axioms  as  have  been  just  noticed,  that  if  equals  be  added 
to  equals  the  wholes  will  be  equal.  And  even  when 
Locke's  assertion,  that  from  these  axioms  no  truths  can 
be  deduced,  is  modified  by  Stewart  and  the  reviewer, 
and  limited  to  geometrical  truths,  it  is  hardly  tenable 
(although,  in  fact,  it  matters  little  to  our  argument 
whether  it  is  or  no).  For  the  greater  part  of  the  Seventh 
Book  of  Euclid's  Elements,  (on  Commensurable  and  In 
commensurable  Quantities,)  and  the  Fifth  Book,  (on 
Proportion,)  depend  upon  these  axioms,  with  the  addi 
tion  only  of  the  definition  or  axiom  (for  it  may  be  stated 
either  way)  which  expresses  the  idea  of  proportionality 
in  numbers.  So  that  the  attempt  to  disprove  the  neces 
sity  and  use  of  axioms,  as  principles  of  reasoning,  fails 
even  when  we  take  those  instances  which  the  opponents 
consider  as  the  more  manifestly  favourable  to  their 

8.  But  perhaps  the  question  may  have  already  sug 
gested  itself  to  the  reader's  mind,  of  what  use  can  it  be 
formally  to  state  such  principles  as  these,  (for  example, 
that  if  equals  be  added  to  equals  the  wholes  are  equal,) 
since,  whether  stated  or  no,  they  will  be  assumed  in  our 
reasoning?  And  how  can  such  principles  be  said  to  be 
necessary,  when  our  proof  proceeds  equally  well  without 
any  reference  to  them?  And  the  answer  is,  that  it  is 
precisely  because  these  are  the  common  principles  of 
reasoning,  which  we  naturally  employ  without  specially 
contemplating  them,  that  they  require  to  be  separated 
from  the  other  steps  and  formally  stated,  when  we 
analyze  the  demonstrations  which  we  have  obtained 
In  every  mental  process  many  principles  are  combined 


and  abbreviated,  and  thus  in  some  measure  concealed 
and  obscured.  In  analyzing  these  processes,  the  combi 
nation  must  be  resolved,  and  the  abbreviation  expanded, 
and  thus  the  appearance  is  presented  of  a  pedantic  and 
superfluous  formality.  But  that  which  is  superfluous  for 
proof,  is  necessary  for  the  analysis  of  proof.  In  order  to 
exhibit  the  conditions  of  demonstration  distinctly,  they 
must  be  exhibited  formally.  In  the  same  manner,  in 
demonstration  we  do  not  usually  express  every  step  in 
the  form  of  a  syllogism,  but  we  see  the  grounds  of  the 
conclusiveness  of  a  demonstration,  by  resolving  it  into 
syllogisms.  Neither  axioms  nor  syllogisms  are  necessary 
for  conviction ;  but  they  are  necessary  to  display  the 
conditions  under  which  conviction  becomes  inevitable. 
The  application  of  a  single  one  of  the  axioms  just  spoken 
of  is  so  minute  a  step  in  the  proof,  that  it  appears  pe 
dantic  to  give  it  a  marked  place ;  but  the  very  essence 
of  demonstration  consists  in  this,  that  it  is  composed  of 
an  indissoluble  succession  of  such  minute  steps.  The 
admirable  circumstance  is,  that  by  the  accumulation  of 
such  apparently  imperceptible  advances,  we  can  in  the 
end  make  so  vast  and  so  sure  a  progress.  The  com 
pleteness  of  the  analysis  of  our  knowledge  appears  in  the 
smallness  of  the  elements  into  which  it  is  thus  resolved. 
The  minuteness  of  any  of  these  elements  of  truth,  of 
axioms  for  instance,  does  not  prevent  their  being  as 
essential  as  others  which  are  more  obvious.  And  any 
attempt  to  assume  one  kind  of  element  only,  when  the 
course  of  our  analysis  brings  before  us  two  or  more 
kinds,  is  altogether  unphilosophical.  Axioms  and  defi 
nitions  are  the  proximate  constituent  principles  of  our 
demonstrations ;  and  the  intimate  bond  which  connects 
together  a  definition  and  an  axiom  on  the  same  subject 
is  not  truly  expressed  by  asserting  the  latter  to  be  de 
rived  from  the  former.  This  bond  of  connexion  exists 


in  the  mind  of  the  reasoner,  in  his  conception  of  that  to 
which  both  definition  and  axiom  refer,  and  consequently 
in  the  general  Fundamental  Idea  of  which  that  concep 
tion  is  a  modification. 


1.  ACCORDING  to  the  views  above  explained,  certain 
of  the  impressions  of  our  senses  convey  to  us  the  per 
ception  of  objects  as  existing  in  space ;  inasmuch  as  by 
the  constitution  of  our  minds  we  cannot  receive  those 
impressions  otherwise  than  in  a  certain  form,  involving 
such  a  manner  of  existence.  But  the  question  deserves 
to  be  asked,  What  are  the  impressions  of  sense  by  which 
we  thus  become  acquainted  with  space  and  its  relations  ? 
And  as  we  have  seen  that  this  idea  of  space  implies  an 
act  of  the  mind  as  well  as  an  impression  on  the  sense, 
what  manifestations  do  we  find  of  this  activity  of  the 
mind,  in  our  observation  of  the  external  world? 

It  is  evident  that  sight  and  touch  are  the  senses  by 
which  the  relations  of  space  are  perceived,  principally  or 
entirely.  It  does  not  appear  that  an  odour,  or  a  feeling 
of  warmth  or  cold,  would,  independently  of  experience, 
suggest  to  us  the  conception  of  a  space  surrounding  us. 
But  when  we  see  objects,  we  see  that  they  are  extended 
and  occupy  space ;  when  we  touch  them,  we  feel  that 
they  are  in  a  space  in  which  we  also  are.  We  have 
before  our  eyes  any  object,  for  instance,  a  board  covered 
with  geometrical  diagrams ;  and  we  distinctly  perceive, 
by  vision,  those  lines  of  which  the  relations  are  the 
subjects  of  our  mathematical  reasoning.  Again,  we  see 
before  us  a  solid  object,  a  cubical  box  for  instance ;  we 
see  that  it  is  within  reach ;  we  stretch  out  the  hand  and 


perceive  by  the  touch  that  it  has  sides,  edges,  corners, 
which  we  had  already  perceived  by  vision. 

2.  Probably  most  persons  do  not  generally  appre 
hend  that  there  is  any  material  difference  in  these  two 
cases ; — that  there  are  any  different  acts  of  mind  con 
cerned  in  perceiving  by  sight  a  mathematical  diagram 
upon  paper,  and  a  solid  cube  lying  on  a  table.  Yet  it  is 
not  difficult  to  show  that,  in  the  latter  case  at  least,  the 
perception  of  the  shape  of  the  object  is  not  immediate. 
A  very  little  attention  teaches  us  that  there  is  an  act  of 
judgment  as  well  as  a  mere  impression  of  sense  requisite, 
in  order  that  we  may  see  any  solid  object.  For  there  is 
no  visible  appearance  which  is  inseparably  connected 
with  solidity.  If  a  picture  of  a  cube  be  rightly  drawn  in 
perspective  and  skilfully  shaded,  the  impression  upon  the 
sense  is  the  same  as  if  it  were  a  real  cube.  The  picture 
may  be  mistaken  for  a  solid  object.  But  it  is  clear  that, 
in  this  case,  the  solidity  is  given  to  the  object  by  an  act 
of  mental  judgment.  All  that  is  seen  is  outline  and 
shade,  figures  and  colours  on  a  flat  board.  The  solid 
angles  and  edges,  the  relation  of  the  faces  of  the  figure 
by  which  they  form  a  cube,  are  matters  of  inference. 
This,  which  is  evident  in  the  case  of  the  pictured  cube,  is 
true  in  all  vision  whatever.  We  see  a  scene  before  us 
on  which  are  various  figures  and  colours,  but  the  eye 
cannot  see  more.  It  sees  length  and  bi'eadth,  but  no 
third  dimension.  In  order  to  know  that  there  are  solids, 
we  must  infer  as  well  as  see.  And  this  we  do  readily 
and  constantly ;  so  familiarly,  indeed,  that  we  do  not 
perceive  the  operation.  Yet  we  may  detect  this  latent 
process  in  many  ways;  for  instance,  by  attending  to 
cases  in  which  the  habit  of  drawing  such  inferences  mis 
leads  us.  Most  persons  have  experienced  this  delusion 
in  looking  at  a  scene  in  a  theatre,  and  especially  that 
kind  of  scene  which  is  called  a  diorama,  when  the 

OF    THE   PERCEPTION    OF    SPACE.  1  1 3 

interior  of  a  building  is  represented.  In  these  cases, 
the  perspective  representations  of  the  various  members 
of  the  architecture  and  decoration  impress  us  almost 
irresistibly  with  the  conviction  that  we  have  before  us  a 
space  of  great  extent  and  complex  form,  instead  of  a  flat 
painted  canvass.  Here,  at  least,  the  space  is  our  own 
creation,  but  yet  here,  it  is  manifestly  created  by  the 
same  act  of  thought  as  if  we  were  really  in  the  palace  or 
the  cathedral  of  which  the  halls  and  aisles  thus  seem  to 
inclose  us.  And  the  act  by  which  we  thus  create  space 
of  three  dimensions  out  of  visible  extent  of  length  and 
breadth,  is  constantly  and  imperceptibly  going  on.  We 
are  perpetually  interpreting  in  this  manner  the  language 
of  the  visible  world.  From  the  appearances  of  things 
which  we  directly  see,  we  are  constantly  inferring  that 
which  we  cannot  directly  see, — their  distance  from  us, 
and  the  position  of  their  parts. 

3.  The  characters  which  we  thus  interpret  are 
various.  They  are,  for  instance,  the  visible  forms, 
colours,  and  shades  of  the  parts,  understood  according 
to  the  maxims  of  perspective ;  (for  of  perspective  every 
one  has  a  practical  knowledge,  as  every  one  has  of 
grammar ;)  the  effort  by  which  we  fix  both  our  eyes  on 
the  same  object,  and  adjust  each  eye  to  distinct  vision ; 
and  the  like.  The  right  interpretation  of  the  informa 
tion  which  such  circumstances  give  us  respecting  the 
true  forms  and  distances  of  things,  is  gradually  learned ; 
the  lesson  being  begun  in  our  earliest  infancy,  and 
inculcated  upon  us  every  hour  during  which  we  use  our 
eyes.  The  completeness  with  which  the  lesson  is  mas 
tered  is  truly  admirable;  for  we  forget  that  our  con 
clusion  is  obtained  indirectly,  and  mistake  a  judgment 
on  evidence  for  an  intuitive  perception.  We  see  the 
breadth  of  the  street,  as  clearly  and  readily  as  we  see 
the  house  on  the  other  side  of  it ;  and  we  see  the  house 
VOL.  i.  w.  p.  1 


to  be  square,  however  obliquely  it  be  presented  to  us. 
This,  however,  by  no  means  throws  any  doubt  or  diffi 
culty  on  the  doctrine  that  in  all  these  cases  we  do  inter 
pret  and  infer.  The  rapidity  of  the  process,  and  the 
unconsciousness  of  the  effort,  are  not  more  remarkable 
in  this  case  than  they  are  when  we  understand  the 
meaning  of  the  speech  which  we  hear,  or  of  the  book 
which  we  read.  In  these  latter  cases  we  merely  hear 
noises  or  see  black  marks ;  but  we  make,  out  of  these 
elements,  thought  and  feeling,  without  being  aware  of 
the  act  by  which  we  do  so.  And  by  an  exactly  similar 
process  we  see  a  variously-coloured  expanse,  and  collect 
from  it  a  space  occupied  by  solid  objects.  In  both 
cases  the  act  of  interpretation  is  become  so  habitual 
that  we  can  hardly  stop  short  at  the  mere  impression 
of  sense. 

4.  But  yet  there  are  various  ways  in  which  we  may 
satisfy  ourselves  that  these  two  parts  of  the  process  of 
seeing  objects  are  distinct.    To  separate  these  operations 
is  precisely  the  task  which  the  artist  has  to  execute  in 
making  a  drawing  of  what  he  sees.     He  has  to  recover 
the  consciousness  of  his  real  and  genuine  sensations,  and 
to  discern  the  lines  of  objects  as  they  appear.     This  at 
first  he  finds  difficult ;  for  he  is  tempted  to  draw  what 
he  knows  of  the  forms  of  visible  objects,  and  not  what 
he  sees :  but  as  he  improves  in  his  art,  he  learns  to  put 
on  paper  what  he  sees  only,  separated  from  what  he 
infers,  in  order  that  thus  the  inference,  and  with  it  a 
conception  like  that  of  the  reality,  may  be  left  to  the 
-spectator.    And  thus  the  natural  process  of  vision  is  the 
habit  of  seeing  that  which  cannot  be  seen ;  and  the  diffi 
culty  of  the  .art  of  drawing  consists  in  learning  not  to 
see  more  than  is  visible. 

5.  But   again ;    even   in   the  simplest    drawing    we 
exhibit    something   which    we    do    not   see.      However 

OF    THE    PERCEPTION    OF   SPACE.  1  1  f) 

slight  is  our  representation  of  objects,  it  contains  some 
thing  which  we  create  for  ourselves.  For  we  draw  an 
outline.  Now  an  outline  has  no  existence  in  nature. 
There  are  no  visible  lines  presented  to  the  eye  by  a 
group  of  figures.  We  separate  each  figure  from  the  rest, 
and  the  boundary  by  which  we  do  this  is  the  outline  of 
the  figure ;  and  the  like  may  be  said  of  each  member  of 
every  figure.  A  painter  of  our  own  times  has  made  this 
remark  in  a  work  upon  his  art*.  "The  effect  which 
natural  objects  produce  upon  our  sense  of  vision  is  that 
of  a  number  of  parts,  or  distinct  masses  of  form  and 
colour,  and  not  of  lines.  But  when  we  endeavour  to 
represent  by  painting  the  objects  which  are  before  us,  or 
which  invention  supplies  to  our  minds,  the  first  and  the 
simplest  means  we  resort  to  is  this  picture,  by  which  we 
separate  the  form  of  each  object  from  those  that  sur 
round  it,  marking  its  boundary,  the  extreme  extent  of 
its  dimensions  in  every  direction,  as  impressed  on  our 
vision :  and  this  is  termed  drawing  its  outline." 

6.  Again,  there  are  other  ways  in  which  we  see  clear 
manifestations  of  the  act  of  thought  by  which  we  assign 
to  the  parts  of  objects  their  relations  in  space,  the  im 
pressions  of  sense  being  merely  subservient  to  this  act. 
If  we  look  at  a  medal  through  a  glass  which  inverts  it, 
we  see  the  figures  upon  it  become  concave  depressions 
instead  of  projecting  convexities ;  for  the  light  which 
illuminates  the  nearer  side  of  the  convexity  will  be  trans 
ferred  to  the  opposite  side  by  the  apparent  inversion  of 
the  medal,  and  will  thus  imply  a  hollow  in  which  the 
side  nearest  the  light  gathers  the  shade.  Here  our  deci 
sion  as  to  which  part  is  nearest  to  us,  has  reference  to 
the  side  from  which  the  light  comes.  In  other  cases 
the  decision  is  more  spontaneous.  If  we  draw  black 
outlines,  such  as  represent  the  edges  of  a  cube  seen 

*   Phillips  On  Painting. 

I  2 


in  perspective,  certain  of  the  lines  will  cross  each  other ; 
and  we  may  make  this  cube  appear  to  assume  two  dif 
ferent  positions,  by  determining  in  our  own  mind  that 
the  lines  which  belong  to  one  end  of  the  cube  shall  be 
understood  to  be  before  or  to  be  behind  those  which 
they  cross.  Here  an  act  of  the  will,  operating  upon  the 
same  sensible  image,  gives  us  two  cubes,  occupying  two 
entirely  different  positions.  Again,  many  persons  may 
have  observed  that  when  a  windmill  in  motion  at  a  dis 
tance  from  us,  (so  that  the  outline  of  the  sails  only  is 
seen,)  stands  obliquely  to  the  eye,  we  may,  by  an  effort 
of  thought,  make  the  obliquity  assume  one  or  the  other 
of  two  positions ;  and  as  we  do  this,  the  sails,  which  in 
one  instance  appear  to  turn  from  right  to  left,  in  the  other 
case  turn  from  left  to  right.  A  person  a  little  familiar 
with  this  mental  effort,  can  invert  the  motion  as  often  as 
he  pleases,  so  long  as  the  conditions  of  form  and  light 
do  not  offer  a  manifest  contradiction  to  either  position. 

Thus  we  have  these  abundant  and  various  manifesta 
tions  of  the  activity  of  the  mind,  in  the  process  by  which 
we  collect  from  vision  the  relations  of  solid  space  of  three 
dimensions.  But  we  must  further  make  some  remarks 
on  the  process  by  which  we  perceive  mere  visible  figure; 
and  also,  on  the  mode  in  which  we  perceive  the  relations 
of  space  by  the  touch ;  and  first,  of  the  latter  subject. 

7.  The  opinion  above  illustrated,  that  our  sight  does 
not  give  us  a  direct  knowledge  of  the  relations  of  solid 
space,  and  that  this  knowledge  is  acquired  only  by  an 
inference  of  the  mind,  was  first  clearly  taught  by  the 
celebrated  Bishop  Berkeley"",  and  is  a  doctrine  now 
generally  assented  to  by  metaphysical  speculators. 

But  does  the  sense  of  touch  give  us  directly  a  know 
ledge  of  space  ?  This  is  a  question  which  has  attracted 
considerable  notice  in  recent  times ;  and  new  light  has 

*    Theory  of  Vision. 

OF  THE  PERCEPTION  OF  SPACE.         117 

been  thrown  upon  it  in  a  degree  which  is  very  remark 
able,  when  we  consider  that  the  philosophy  of  perception 
has  been  a  prominent  subject  of  inquiry  from  the  earliest 
times.  Two  philosophers,  advancing  to  this  inquiry  from 
different  sides,  the  one  a  metaphysician,  the  other  a  phy 
siologist,  have  independently  arrived  at  the  conviction 
that  the  long  current  opinion,  according  to  which  we 
acquire  a  knowledge  of  space  by  the  sense  of  touch,  is 
erroneous.  And  the  doctrine  which  they  teach  instead 
of  the  ancient  errour,  has  a  very  important  bearing  upon 
the  principle  which  we  are  endeavouring  to  establish,— 
that  our  knowledge  of  space  and  its  properties  is  derived 
rather  from  the  active  operations  than  from  the  passive 
impressions  of  the  percipient  mind. 

Undoubtedly  the  persuasion  that  we  acquire  a  know 
ledge  of  form  by  the  touch  is  very  obviously  suggested 
by  our  common  habits.  If  we  wish  to  know  the  form  of 
any  body  in  the  dark,  or  to  correct  the  impressions  con 
veyed  by  sight,  when  we  suspect  them  to  be  false,  we 
have  only,  it  seems  to  us,  at  least  at  first,  to  stretch  forth 
the  hand  and  touch  the  object ;  and  we  learn  its  shape 
with  no  chance  of  error.  In  these  cases,  form  appears 
to  be  as  immediate  a  perception  of  the  sense  of  touch, 
as  colour  is  of  the  sense  of  sight. 

8.  But  is  this  perception  really  the  result  of  the 
passive  sense  of  touch  merely  ?  Against  such  an  opinion 
Dr.  Brown,  the  metaphysician  of  whom  I  speak,  urges* 
that  the  feeling  of  touch  alone,  when  any  object  is  ap 
plied  to  the  hand,  or  any  other  part  of  the  body,  can  no 
more  convey  the  conception  of  form  or  extension,  than 
the  sensation  of  an  odour  or  a  taste  can  do,  except  we 
have  already  some  knowledge  of  the  relative  position  of 
the  parts  of  our  bodies;  that  is,  except  we  are  already  in 
possession  of  an  idea  of  space,  and  have,  in  our  minds, 
*  Lectures,  Vol.  i.  p.  459,  (182-1). 


referred  our  limbs  to  their  positions;  which  is  to  sup 
pose  the  conception  of  form  already  acquired. 

9.  By  what  faculty  then  do  we  originally  acquire  our 
conceptions  of  the  relations  of  position  ?  Brown  answers 
by  the  muscular  sense ;  that  is,  by  the  conscious  exer 
tions  of  the  various  muscles  by  which  we  move  our  limbs. 
When  we  feel  out  the  form  and  position  of  bodies  by 
the  hand,  our  knowledge  is  acquired,  not  by  the  mere 
touch  of  the  body,  but  by  perceiving  the  course  the 
fingers  must  take  in  order  to'  follow  the  surface  of  the 
body,  or  to  pass  from  one  body  to  another.  We  are 
conscious  of  the  slightest  of  the  volitions  by  which  we 
thus  feel  out  form  and  place  ;  we  know  whether  we  move 
the  finger  to  the  right  or  left,  up  or  down,  to  us  or  from 
us,  through  a  large  or  a  small  space ;  and  all  these  con 
scious  acts  are  bound  together  and  regulated  in  our 
minds  by  an  idea,  of  an  extended  space  in  which  they  are 
performed.  That  this  idea  of  space  is  not  borrowed  from 
the  sight,  and  transferred  to  the  muscular  feelings  by 
habit,  is  evident.  For  a  man  born  blind  can  feel  out  his 
way  with  his  staff,  and  has  his  conceptions  of  position 
determined  by  the  conditions  of  space,  no  less  than  one 
who  has  the  use  of  his  eyes.  And  the  muscular  con 
sciousness  which  reveals  to  us  the  position  of  objects  and 
parts  of  objects,  when  we  feel  them  out  by  means  of  the 
hand,  shews  itself  in  a  thousand  other  ways,  and  in  all 
our  limbs :  for  our  habits  of  standing,  walking,  and  all 
other  attitudes  and  motions,  are  regulated  by  our  feeling 
of  our  position  and  that  of  surrounding  objects.  And 
thus,  we  cannot  touch  any  object  without  learning  some 
thing  respecting  its  position ;  not  that  the  sense  of 
touch  directly  conveys  such  knowledge ;  but  we  have 
already  learnt,  from  the  muscular  sense,  constantly 
exercised,  the  position  of  the  limb  which  the  object  thus 

OF    THE    PERCEPTION    OF    SPACE.  1  1 9 

10.  The  justice  of  this  distinction  will,  I  think,  be 
assented  to  by  all  persons  who  attend  steadily  to  the 
process  itself,  and  might  be  maintained  by  many  forcible 
reasons.     Perhaps  one  of  the  most  striking  evidences  in 
its  favour  is  that,  as  I  have  already  intimated,  it  is  the 
opinion  to  which  another  distinguished  philosopher,  Sir 
Charles  Bell,  has  been  led,  reasoning  entirely  upon  phy 
siological    principles.     From  his  researches  it  resulted 
that  besides  the  nerves  which  convey  the  impulse  of  the 
will  from  the  brain  to  the  muscle,  by  which  every  motion 
of  our  limbs  is  produced,  there  is  another  set  of  nerves 
which  carry  back  to  the  brain  a  sense  of  the  condition 
of  the  muscle,  and  thus  regulate  its  activity ;  and  give  us 
the  consciousness  of  our  position  and  relation  to  sur 
rounding  objects.     The  motion  of  the  hand  and  fingers, 
or  the  consciousness  of  this  motion,  must  be  combined 
with  the  sense  of  touch  properly  so  called,  in  order  to 
make  an  inlet  to  the  knowledge  of  such  relations.     This 
consciousness  of  muscular  exertion,  which  he  has  called  a 
sixth  sense"",  is  our  guide,  Sir  C.  Bell  shows,  in  the  com 
mon  practical  government  of  our  motions ;  and  he  states 
that  having  given  this  explanation  of  perception  as  a 
physiological  doctrine,  he  had  afterwards  with  satisfac 
tion  seen  it  confirmed  by  Dr.  Brown's  speculations. 

11.  Thus  it  appears  that  our  consciousness  of  the 
relations  of  space  is  inseparably  and  fundamentally  con 
nected  with  our  own  actions  in  space.    We  perceive  only 
while  we  act ;  our  sensations  require  to  be  interpreted  by 
our  volitions.     The  apprehension  of  extension  and  figure 
is  far  from  being  a  process  in  which  we  are  inert  and 
passive.     We  draw  lines  with  our  fingers ;  we  construct 
surfaces  by  curving  our  hands;  we  generate  spaces  by  the 
motion  of  our  arms.     When  the  geometer  bids  us  form 
lines,  or  surfaces,  or  solids  by  motion,   he  intends  his 

*  Bridgetvater  Treatise,  p.  195.    Phil.  Trans.  1K2<>,  I't.  11.,  p.  H>7- 


injunction  to  be  taken  as  hypothetical  only  ;  we  need  only 
conceive  such  motions.  But  yet  this  hypothesis  repre 
sents  truly  the  origin  of  our  knowledge ;  we  perceive 
spaces  by  motion  at  first,  as  we  conceive  spaces  by  motion 
afterwards : — or  if  not  always  by  actual  motion,  at  least 
by  potential.  If  we  perceive  the  length  of  a  staff  by 
holding  its  two  ends  in  our  two  hands  without  running 
the  finger  along  it,  this  is  because  by  habitual  motion  we 
have  already  acquired  a  measure  of  the  distance  of  our 
hands  in  any  attitude  of  which  we  are  conscious.  Even 
in  the  simplest  case,  our  perceptions  are  derived  not  from 
the  touch,  but  from  the  sixth  sense ;  and  this  sixth  sense 
at  least,  whatever  may  be  the  case  with  the  other  five, 
implies  an  active  mind  along  with  the  passive  sense. 

12.  Upon  attentive  consideration,  it  will  be  clear 
that  a  large  portion  of  the  perceptions  respecting  space 
which  appear  at  first  to  be  obtained  by  sight  alone,  are, 
in  fact,  acquired  by  means  of  this  sixth  sense.  Thus  we 
consider  the  visible  sky  as  a  single  surface  surrounding 
us  and  returning  into  itself,  and  thus  forming  a  hemi 
sphere.  But  such  a  mode  of  conceiving  an  object  of  vision 
could  never  have  occurred  to  us,  if  we  had  not  been  able 
to  turn  our  heads,  to  follow  this  surface,  to  pursue  it  till 
we  find  it  returning  into  itself.  And  when  we  have  done 
this,  we  necessarily  present  it  to  ourselves  as  a  concave 
inclosure  within  which  we  are.  The  sense  of  sight  alone, 
without  the  power  of  muscular  motion,  could  not  have 
led  us  to  view  the  sky  as  a  vault  or  hemisphere.  Under 
such  circumstances,  we  should  have  perceived  only  what 
was  presented  to  the  eye  in  one  position ;  and  if  dif 
ferent  appearances  had  been  presented  in  succession,  we 
could  not  have  connected  them  as  parts  of  the  same 
picture,  for  want  of  any  perception  of  their  relative  posi 
tion.  They  would  have  been  so  many  detached  and 
incoherent  visual  sensations.  The  muscular  sense  con- 

OF  THE  PERCEPTION  OF  SPACE.         121 

nccts  their  parts  into  a  whole,  making  them  to  be  only 
different  portions  of  one  universal  scene  *. 

13.  These  considerations  point  out  the  fallacy  of  a 
very  curious  representation  made  by  Dr.  Reid,  of  the 
convictions  to  which  man  would  be  led,  if  he  possessed 
vision  without  the  sense  of  touch.  To  illustrate  this  sub 
ject,  Reid  uses  the  fiction  of  a  nation  whom  he  terms  the 
Idomenians,  who  have  no  sense  except  that  of  sight.  He 
describes  their  notions  of  the  relations  of  space  as  being 
entirely  different  from  ours.  The  axioms  of  their  geome 
try  are  quite  contradictory  to  our  axioms.  For  example, 
it  is  held  to  be  self-evident  among  them  that  two  straight 
lines  which  intersect  each  other  once,  must  intersect  a 
second  time ;  that  the  three  angles  of  any  triangle  are 
greater  than  two  right  angles;  and  the  like.  These 
paradoxes  are  obtained  by  tracing  the  relations  of  lines 
on  the  surface  of  a  concave  sphere,  which  surrounds  the 
spectator,  and  on  which  all  visible  appearances  may  be 
supposed  to  be  presented  to  him.  But  from  what  is  said 
above  it  appears  that  the  notion  of  such  a  sphere,  and 
such  a  connexion  of  visible  objects  which  are  seen  in  dif 
ferent  directions,  cannot  be  arrived  at  by  sight  alone. 

*  It  has  been  objected  to  this  view,  that  we  might  obtain  a  con 
ception  of  the  sky  as  a  hemisphere,  by  being  ourselves  turned  round,  (as 
on  a  music-stool,  for  instance,)  and  thus  seeing  in  succession  all  parts  of 
the  sky.  But  this  assertion  I  conceive  to  be  erroneous.  By  being  thus 
turned  round,  we  should  see  a  number  of  pictures  which  we  should  put 
together  as  parts  of  a  plane  picture ;  and  when  we  came  round  to  the 
original  point,  we  should  have  no  possible  means  of  deciding  that  it 
mas  the  same  point :  it  would  appear  only  as  a  repetition  of  the  pic 
ture.  That  sight,  of  itself,  can  give  us  only  a  plane  picture,  the  doctrine 
of  Berkeley,  appears  to  be  indisputable ;  and,  no  less  so,  the  doctrine 
that  it  is  the  consciousness  of  our  own  action  in  space  which  puts  toge 
ther  these  pictures  so  that  they  cover  the  surface  of  a  solid  body.  We 
can  see  length  and  breadth  with  our  eyes,  but  we  must  thrust  out  our 
arm  towards  the  flat  surface,  in  order  that  we  may,  in  our  thoughts, 
combine  a  third  dimension  with  the  other  two. 


When  the  spectator  combines  in  his  conception  the  rela 
tions  of  long-drawn  lines  and  large  figures,  as  he  sees 
them  by  turning  his  head  to  the  right  and  to  the  left, 
upwards  and  downwards,  he  ceases  to  be  an  Idomenian. 
And  thus  our  conceptions  of  the  properties  of  space,  de 
rived  through  the  exercise  of  one  mode  of  perception, 
are  not  at  variance  with  those  obtained  in  another  way ; 
but  all  such  conceptions,  however  produced  or  suggested, 
are  in  harmony  with  each  other ;  being,  as  has  already 
been  said,  only  different  aspects  of  the  same  idea. 

14.  If  our  perceptions   of  the   position  of  objects 
around  us  do  not  depend  on  the  sense  of  vision  alone, 
but  on  the  muscular  feeling  brought  into  play  when  we 
turn  our  head,  it  will  obviously  follow  that  the  same  is 
true  when  we  turn  the  eye  instead  of  the  head.     And 
thus  we  may  learn  the  form  of  objects,  not  by  looking 
at  them  with  a  fixed  gaze,  but  by  following  the  boundary 
of  them  with  the  eye.     While  the  head  is  held  perfectly 
still,  the  eye  can  rove  along  the  outlines  of  visible  ob 
jects,  scrutinize  each  point  in  succession,  and  leap  from 
one  point  to  another ;  each  such  act  being  accompanied 
by  a  muscular  consciousness  which  makes  us  aware  of 
the  direction  in  which  the  look  is  travelling.     And  we 
may  thus  gather  information  concerning  the  figures  and 
places  which  we  trace  out  with  the  visual  ray,  as  the 
blind  man  learns  the  forms  of  things  which  he  traces  out 
with  his  staff,  being  conscious  of  the  motions  of  his  hand. 

15.  This  view  of  the  mode  in  which  the  eye  per 
ceives  position,  which  is  thus  supported  by  the  analogy 
of  other  members  employed  for  the  same  purpose,  is 
further  confirmed  by  Sir  Charles  Bell  by  physiological 
reasons.    He  teaches  us  that"""  when  an  object  is  seen  we 
employ  two  senses:  there  is  an  impression  on  the  retina; 
but  we  receive  also  the  idea  of  position  or  relation  in 

*  Phil.  Trans.,  1823.     On  the  Motions  of  the  Eye. 

OF  THE  PERCEPTION  OF  SPACE.        1  ±3 

space,  which  it  is  not  the  office  of  the  retina  to  give,  by 
our  consciousness  of  the  efforts  of  the  voluntary  muscles 
of  the  eve :  and  he  has  traced  in  detail  the  course  of  the 


nerves  by  which  these  muscles  convey  their  information. 
The  constant  searching  motion  of  the  eye,  as  he  terms 
it*,  is  the  means  by  which  we  become  aware  of  the 
position  of  objects  about  us. 

16.  It  is  not  to  our  present  purpose  to  follow'  the 
physiology  of  this  subject ;  but  we  may  notice  that  Sir 
C.  Bell  has  examined  the  special  circumstances  which 
belong  to  this  operation  of  the  eye.  We  learn  from  him 
that  the  particular  point  of  the  eye  which  thus  traces  the 
forms  of  visible  objects  is  a  part  of  the  retina  which  has 
been  termed  the  sensible  spot;  being  that  part  which  is 
most  distinctly  sensible  to  the  impressions  of  light  and 
colour.  This  part,  indeed,  is  not  a  spot  of  definite  size  and 
form,  for  it  appears  that  proceeding  from  a  certain  point 
of  the  retina,  the  distinct  sensibility  diminishes  on  every 
side  by  degrees.  And  the  searching  motion  of  the  eye 
arises  from  the  desire  which  we  instinctively  feel  of  re 
ceiving  upon  the  sensible  spot  the  image  of  the  object 
to  which  the  attention  is  directed.  We  are  uneasy  and 

*  Bridgtwater  Treatise,  p.  282.  I  have  adopted,  in  writing  the 
ahove,  the  views  and  expressions  of  Sir  Charles  Bell.  The  essential 
part  of  the  doctrine  there  presented  is,  that  the  eye  constantly  makes 
efforts  to  turn,  so  that  the  image  of  an  object  to  which  our  attention  is 
drawn,  shall  fall  upon  a  certain  particular  point  of  the  retina;  and  that 
when  the  image  falls  upon  any  other  point,  the  eye  turns  away  from 
this  oblique  into  the  direct  position.  Other  writers  have  maintained 
that  the  eye  thus  turns,  not  because  the  point  on  which  the  image  falls 
in  direct  vision  is  the  moxt  .sensible  point,  but  that  it  is  the  point  of 
greatest  distinctness  of  vision.  They  urge  that  a  small  star,  which  dis 
appears  when  the  eye  is  turned  full  upon  it,  may  often  be  seen  by 
looking  a  little  away  from  it :  and  hence,  they  infer  that  the  parts  of 
the  retina  removed  from  the  spot  of  direct  vision,  are  more  sensible  than 
it  is.  The  facts  are  very  curious,  however  they  lx)  explained,  but  they 
do  not  disturb  the  doetrinc  delivered  in  the  text. 


impatient  till  the  eye  is  turned  so  that  this  is  effected. 
And  as  our  attention  is  transferred  from  point  to  point 
of  the  scene  before  us,  the  eye,  and  this  point  of  the  eye 
in  particular,  travel  along  with  the  thoughts ;  and  the 
muscular  sense,  which  tells  us  of  these  movements  of 
the  organ  of  vision,  conveys  to  us  a  knowledge  of  the 
forms  and  places  which  we  thus  successively  survey. 

17.  How  much  of  activity  there  is  in  the  process  by 
which  we  perceive  the  outlines  of  objects  appears  further 
from  the  language  by  which  we  describe  their  forms. 
We  apply  to  them  not  merely  adjectives  of  form,  but 
verbs  of  motion.  An  abrupt  hill  starts  out  of  the  plain ; 
a  beautiful  figure  has  a  gliding  outline.  We  have 

The  windy  summit,  wild  and  high, 
Roughly  rushing  on  the  sky. 

These  terms  express  the  course  of  the  eye  as  it  follows 
the  lines  by  which  such  forms  are  bounded  and  marked. 
In  like  manner  another  modern  poet*  says  of  Soracte, 
that  it 

From  out  the  plain 

Heaves  like  a  long-swept  wave  about  to  break, 
And  on  the  curl  hangs  pausing. 

Thus  the  muscular  sense,  which  is  inseparably  con 
nected  with  an  act  originating  in  our  own  mind,  not  only 
gives  us  all  that  portion  of  our  perceptions  of  space  in 
which  wre  use  the  sense  of  touch,  but  also,  at  least  in  a 
great  measure,  another  large  portion  of  such  perceptions, 
in  which  we  employ  the  sense  of  sight.  As  we  have 
before  seen  that  our  knowledge  of  solid  space  and  its 
properties  is  not  conceivable  in  any  other  way  than  as 
the  result  of  a  mental  act,  governed  by  conditions  depend 
ing  on  its  own  nature ;  so  it  nowT  appears  that  our  per 
ceptions  of  visible  figure  are  not  obtained  without  an  act 
performed  under  the  same  conditions.  The  sensations 
of  touch  and  sight  are  subordinated  to  an  idea  which  is 

*  Byron,  Ch.  Har.  vi.,  st.  75. 

OF    THE    PERCEPTION     OF    SPACE.  \'2i) 

the  basis  of  our  speculative  knowledge  concerning  space 
and  its  relations ;  and  this  same  idea  is  disclosed  to  our 
consciousness  by  its  practically  regulating  our  inter 
course  with  the  external  world. 

By  considerations  such  as  have  been  adduced  and 
referred  to,  it  is  proved  beyond  doubt,  that  in  a  great 
number  of  cases  our  knowledge  of  form  and  position  is 
acquired  from  the  muscular  sense,  and  not  from  sight 
directly: — for  instance,  in  all  cases  in  which  we  have 
before  us  objects  so  large  and  prospects  so  extensive 
that  we  cannot  see  the  whole  of  them  in  one  position  of 
the  eye*. 

We  now  quit  the  consideration  of  the  properties  of 
Space,  and  consider  the  Idea  of  Time. 

OF    THE    IDEA    OF    TIME. 

1.    RESPECTING   the   Idea   of  Time,  we  may   make 
several  of  the  same  remarks  which  we  made  concerning 

*  The  expression  in  the  first  edition  was  "  large  objects  and  exten 
sive  spaces."  In  the  text  as  now  given,  I  state  a  definite  size  and 
extent,  within  which  the  sight  by  itself  can  judge  of  position  and  figure. 

The  doctrine  that  we  require  the  assistance  of  the  muscular  sense  to 
enable  us  to  perceive  space  of  three  dimensions,  is  not  at  all  inconsistent 
with  this  other  doctrine,  that  within  the  space  which  is  seen  by  the 
fixed  eye,  we  perceive  the  relative  positions  of  points  directly  by  vision, 
and  that,  consequently,  we  have  a  perception  of  visible  .figure. 

Sir  Charles  Bell  has  said,  (Phil.  Trans.  1823,  p.  181,)  "It  appears 
to  me  that  the  utmost  ingenuity  will  be  at  a  loss  to  devise  an  explana 
tion  of  that  power  by  which  the  eye  becomes  acquainted  with  the 
position  and  relation  of  objects,  if  the  sense  of  muscular  activity  be 
excluded  which  accompanies  the  motion  of  the  eyeball."  But  surely  we 
should  have  no  difficulty  in  perceiving  the  relation  of  the  sides  and 
angles  of  a  small  triangle,  placed  before  the  eye,  even  if  the  muscles  of 
the  eyeball  were  severed.  This  subject  is  resumed  13.  iv.  c.  ii.  sect.  1 1. 


the  idea  of  space,  in  order  to  shew  that  it  is  not  bor 
rowed  from  experience,  but  is  a  bond  of  connexion 
among  the  impressions  of  sense,  derived  from  a  peculiar 
activity  of  the  mind,  and  forming  a  foundation  both  of 
our  experience  and  of  our  speculative  knowledge. 

Time  is  not  a  notion  obtained  by  experience.  Expe 
rience,  that  is,  the  impressions  of  sense  and  our  con 
sciousness  of  our  thoughts,  gives  us  various  percep 
tions;  and  different  successive  perceptions  considered 
together  exemplify  the  notion  of  change.  But  this  very 
connexion  of  different  perceptions, — this  successiveness, 
—presupposes  that  the  perceptions  exist  in  time.  That 
things  happen  either  together,  or  one  after  the  other,  is 
intelligible  only  by  assuming  time  as  the  condition  under 
which  they  are  presented  to  us. 

Thus  time  is  a  necessary  condition  in  the  presentation 
of  all  occurrences  to  our  minds.  We  cannot  conceive 
this  condition  to  be  taken  away.  We  can  conceive 
time  to  go  on  while  nothing  happens  in  it ;  but  we  can 
not  conceive  anything  to  happen  while  time  does  not 
go  on. 

It  is  clear  from  this  that  time  is  not  an  impression 
derived  from  experience,  in  the  same  manner  in  which 
we  derive  from  experience  our  information  concerning 
the  objects  which  exist,  and  the  occurrences  which  take 
place  in  time.  The  objects  of  experience  can  easily  be 
conceived  to  be,  or  not  to  be : — to  be  absent  as  well  as 
present.  Time  always  is,  and  always  is  present,  and 
even  in  our  thoughts  we  cannot  form  the  contrary  sup 

2.  Thus  time  is  something  distinct  from  the  matter 
or  substance  of  our  experience,  and  may  be  considered 
as  a  necessary  form  which  that  matter  (the  experience  of 
change)  must  assume,  in  order  to  be  an  object  of  con 
templation  to  the  mind.  Time  is  one  of  the  necessary 

OF    THE    IDEA    OF    TIME.  \'27 

conditions  under  which  we  apprehend  the  information 
which  our  senses  and  consciousness  give  us.  By  con 
sidering  time  as  a  form  which  belongs  to  our  power  of 
apprehending  occurrences  and  changes,  and  under  which 
alone  all  such  experience  can  be  accepted  by  the  mind, 
we  explain  the  necessity,  which  we  find  to  exist,  of  con 
ceiving  all  such  changes  as  happening  in  time ;  and  we 
thus  see  that  time  is  not  a  property  perceived  as  existing 
in  objects,  or  as  conveyed  to  us  by  our  senses ;  but  a  con 
dition  impressed  upon  our  knowledge  by  the  constitution 
of  the  mind  itself;  involving  an  act  of  thought  as  well  as 
an  impression  of  sense. 

3.  We  showed  that  space  is  an  idea  of  the  mind,  or 
form  of  our  perceiving  power,  independent  of  experience, 
by  pointing  out  that  we  possess  necessary  and  universal 
truths  concerning  the  relations  of  space,    which   could 
never  be  given  by  means  of  experience ;    but  of  which 
the  necessity  is  readily  conceivable,  if  we  suppose  them 
to  have  for  their  basis  the  constitution   of  the   mind. 
There  exist  also  respecting  number,  many  truths  abso 
lutely  necessary,  entirely  independent  of  experience  and 
anterior  to  it ;  and  so  far  as  the  conception  of  number 
depends  upon  the  idea  of  time,  the  same  argument  might 
be  used  to  show  that  the  idea  of  time  is  not  derived  from 
experience,  but  is  a  result  of  the  native  activity  of  the 
mind :  but  we  shall  defer  all  views  of  this  kind  till  we 
come  to  the  consideration  of  Number. 

4.  Some  persons  have  supposed  that  we  obtain  the 
notion  of  time  from  the  perception  of  motion.     But  it 
is  clear  that  the  perception  of  motion,  that  is,  change  of 
place,   presupposes  the  conception  of  time,  and  is  not 
capable  of  being  presented  to  the  mind  in  any  other  way. 
If  we  contemplate  the  same  body  as  being  in  different 
places  at  different  times,  and  connect  these  observations, 
we  have  the  conception  of  motion,  which  thus  presup- 


poses  the  necessary  conditions  that  existence  in  time 
implies.  And  thus  we  see  that  it  is  possible  there  should 
be  necessary  truths  concerning  all  motion,  and  conse 
quently,  concerning  those  motions  which  are  the  objects 
of  experience ;  but  that  the  source  of  this  necessity  is  the 
Ideas  of  time  and  space,  which,  being  universal  conditions 
of  knowledge  residing  in  the  mind,  afford  a  foundation 
for  necessary  truths. 


1.  THE  Idea  of  Time,  like  the  Idea  of  Space,  offers  to 
our  notice  some  characters  which  do  not  belong  to  our 
fundamental  ideas  generally,  but  which  are  deserving  of 
remark.  These  characters  are,  in  some  respects,  closely 
similar  with  regard  to  time  and  to  space,  while,  in  other 
respects,  the  peculiarities  of  these  two  ideas  are  widely 
different.  We  shall  point  out  some  of  these  characters. 

Time  is  not  a  general  abstract  notion  collected  from 
experience ;  as,  for  example,  a  certain  general  concep 
tion  of  the  relations  of  things.  For  we  do  not  consider 
particular  times  as  examples  of  Time  in  general,  (as  we 
consider  particular  causes  to  be  examples  of  Cause,)  but 
we  conceive  all  particular  times  to  be  parts  of  a  single 
and  endless  Time.  This  continually-flowing  and  endless 
time  is  what  offers  itself  to  us  when  we  contemplate  any 
series  of  occurrences.  All  actual  and  possible  times 
exist  as  Parts,  in  this  original  and  general  Time.  And 
since  all  particular  times  are  considered  as  derivable 
from  time  in  general,  it  is  manifest  that  the  notion  of 
time  in  general  cannot  be  derived  from  the  notions  of 
particular  times.  The  notion  of  time  in  general  is  there- 

SOME   PECULIARITIES    OF    THE    IDEA    OF    TIME.        120 

fore  not  a  general  conception   gathered   from   experi 

2.  Time  is  infinite.      Since  all  actual  and  possible 
times  exist  in  the  general  course  of  time,  this  general 
time    must   be   infinite.     All  limitation  merely  divides, 
and  does  not  terminate,  the   extent   of  absolute   time. 
Time  has  no  beginning  and  no  end ;  but  the  beginning 
and  the  end  of  every  other  existence  takes  place  in  it. 

3.  Time,  like  space,  is  not  only  a  form  of  perception, 
but    of  intuition.     We  contemplate    events   as  taking 
place  in  time.     We  consider  its  parts  as  added  to  one 
another,  and  events  as  filling  a  larger  or  smaller  extent 
of  such  parts.     The  time  which  any  event  takes  up  is 
the  sum  of  all  such  parts,  and  the  relation  of  the  same 
to  time  is  fully  understood  when  we  can  clearly  see  what 
portions  of  time    it    occupies,   and    what    it   does    not. 
Thus   the  relation   of  known    occurrences   to    time   is 
perceived  by  intuition ;  and  time  is  a  form  of  intuition 
of  the  external  world. 

4.  Time  is  conceived  as  a  quantity  of  one  dimension  ; 
it  has  great  analogy  with  a  line,  but  none  at  all  with  a 
surface  or  solid.     Time  may  be  considered  as  consisting 
of  a  series  of  instants,  which  are  before  and  after  one 
another;  and  they  have  no  other  relation  than  this,  of 
before  and  after.     Just  the  same  would  be  the  case  with 
a  series  of  points  taken  along  a  line ;   each  would  be 
after  those  on  one  side  of  it,  and  before  those  on  another. 
Indeed  the  analogy   between  time,   and   space    of  one 
dimension,  is  so  close,  that  the  same  terms  are  applied  to 
both  ideas,  and  we  hardly  know  to  which  they  originally 
belong.     Times  and  lines  are  alike  called  long  and  short ; 
we  speak  of  the  beginning  and  end  of  a  line ;  of  a  point 
of  time,  and  of  the  limits  of  a  portion  of  duration. 

5.  But,  as  has  been  said,  there  is  nothing  in  time 
which  corresponds  to  more  than  one  dimension  in  space, 

VOL.  i.    \v.  r.  K 


and  hence  nothing  which  has  any  obvious  analogy  with 
figure  Time  resembles  a  line  indefinitely  extended  both 
ways ;  all  partial  times  are  portions  of  this  line ;  and  no 
mode  of  conceiving  time  suggests  to  us  a  line  making 
any  angle  with  the  original  line,  or  any  other  combina 
tion  which  might  give  rise  to  figures  of  any  kind.  The 
analogy  between  time  and  space,  which  in  many  circum 
stances  is  so  clear,  here  disappears  altogether.  Spaces 
of  two  and  of  three  dimensions,  planes  and  solids,  have 
nothing  to  which  we  can  compare  them  in  the  concep 
tions  arising  out  of  time. 

6.  As  figure  is  a  conception  solely  appropriate  to 
space,  there  is  also  a  conception  which  peculiarly  belongs 
to  time,  namely,  the  conception  of  recurrence  of  times 
similarly  marked ;  or,  as  it  may  be  termed,  rhythm, 
using  this  word  in  a  general  sense.  The  term  rhythm 
is  most  commonly  used  to  designate  the  recurrence  of 
times  marked  by  the  syllables  of  a  verse,  or  the  notes  of 
a  melody :  but  it  is  easy  to  see  that  the  general  concep 
tion  of  such  a  recurrence  does  not  depend  on  the  mode 
in  which  it  is  impressed  upon  the  sense.  The  forms  of 
such  recurrence  are  innumerable.  Thus  in  such  a  line  as 

Quadnipedante  putrem  sonitu  quatit  lingula  campum, 

we  have  alternately  one  long  or  forcible  syllable,  and 
two  short  or  light  ones,  recurring  over  and  over.  In 
like  manner  in  our  own  language,  in  the  line 

At  the  close  of  the  day  when  the  hamlet  is  still, 

we  have  two  light  and  one  strong  syllable  repeated  four 
times  over.  Such  repetition  is  the  essence  of  versification. 
The  same  kind  of  rhythm  is  one  of  the  main  elements  of 
music,  with  this  difference  only,  that  in  music  the  forcible 
syllables  are  made  so  for  the  purposes  of  rhythm  by 
their  length  only  or  principally  ;  for  example,  if  either  of 
the  above  lines  were  imitated  by  a  melody  in  the  most 


simple  and  obvious  manner,  each  strong  syllable  would 
occupy  exactly  twice  as  much  time  as  two  of  the  weaker 
ones.  Something  very  analogous  to  such  rhythm  may 
be  traced  in  other  parts  of  poetry  and  art,  which  we  need 
not  here  dwell  upon.  But  in  reference  to  our  present 
subject,  we  may  remark  that  by  the  introduction  of  such 
rhythm,  the  flow  of  time,  which  appears  otherwise  so 
perfectly  simple  and  homogeneous,  admits  of  an  infinite 
number  of  varied  yet  regular  modes  of  progress.  All 
the  kinds  of  versification  which  occur  in  all  languages, 
and  the  still  more  varied  forms  of  recurrence  of  notes  of 
different  lengths,  which  are  heard  in  all  the  varied  strains 
of  melodies,  are  only  examples  of  such  modifications,  or 
configurations  as  we  may  call  them,  of  time.  They  in 
volve  relations  of  various  portions  of  time,  as  figures 
involve  relations  of  various  portions  of  space.  But  yet 
the  analogy  between  rhythm  and  figure  is  by  no  means 
very  close ;  for  in  rhythm  we  have  relations  of  quantity 
alone  in  the  parts  of  time,  whereas  in  figure  we  have  re 
lations  not  only  of  quantity,  but  of  a  kind  altogether 
different, — namely,  of  position.  On  the  other  hand,  a 
repetition  of  similar  elements,  which  does  not  necessarily 
occur  in  figures,  is  quite  essential  in  order  to  impress 
upon  us  that  measured  progress  of  time  of  which  we  here 
speak.  And  thus  the  ideas  of  time  and  space  have  each 
its  peculiar  and  exclusive  relations ;  position  and  figure 
belonging  only  to  space,  while  repetition  and  rhythm  are 
appropriate  to  time. 

7.  One  of  the  simplest  forms  of  recurrence  is  alter 
nation,  as  when  we  have  alternate  strong  and  slight  syl 
lables.  For  instance,— 

Awake,  arise,  or  be  for  £ver  fdll'n. 

Or    without    any   subordination,   as    when    we   reckon 
numbers,  and  call  them   in  succession,  odd,  even, 

K  a 


8.  But  the  simplest  of  all  forms  of  recurrence  is  that 
which  has  no  variety ; — in  which  a  series  of  units,  each 
considered  as  exactly  similar  to  the  rest,  succeed  each 
other ;  as  one,  one,  one,  and  so  on.     In  this  case,  how 
ever,  we  are  led  to  consider  each  unit  with  reference  to 
all  that  have  preceded ;  and  thus  the  series  one,  one,  one, 
and  so  forth,  becomes  one,  two,  three,  four,  Jive,  and  so 
on  ;  a  series  with  which  all  are  familiar,  and  which  may 
be  continued  without  limit. 

We  thus  collect  from  that  repetition  of  which  time 
admits,  the  conception  of  Number. 

9.  The  relations  of  position  and  figure  are  the  sub 
ject  of  the  science  of  geometry ;    and  are,  as  we  have 
already  said,  traced  into  a  very  remarkable  and  extensive 
body  of  truths,  which  rests  for  its  foundations  on  axioms 
involved  in  the  Idea  of  Space.     There  is,  in  like  manner, 
a  science  of  great  complexity  and  extent,  which  has  its 
foundation  in  the  Idea  of  Time.     But  this  science,  as  it 
is  usually  pursued,  applies  only  to  the  conception  of  Num 
ber,  which  is,  as  we  have  said,  the  simplest  result    of 
repetition.     This  science  is   Theoretical  Arithmetic,  or 
the  speculative  doctrine  of  the  properties  and  relations 
of  numbers ;  and  we  must  say  a  few  words  concerning 
the  principles  which  it  is  requisite  to  assume  as  the  basis 
of  this  science. 


I.  TPIE  foundations  of  our  speculative  knowledge  of 
the  relations  and  properties  of  Number,  as  well  as  of 
Space,  are  contained  in  the  mode  in  which  we  represent  to 
ourselves  the  magnitudes  which  are  the  subjects  of  our 
reasonings.  To  express  these  foundations  in  axioms  in  the 

OF    THE    AXIOMS    WHICH    RELATE    TO    NUMBKK.      133 

case  of  number,  is  a  matter  requiring  some  consideration, 
for  the  same  reason  as  in  the  case  of  geometry ;  that  is, 
because  these  axioms  are  principles  which  we  assume  as 
true,  without  being  aware  that  we  have  made  any  assump 
tion  ;  and  we  cannot,  without  careful  scrutiny,  determine 
when  we  have  stated,  in  the  form  of  axioms,  all  that  is 
necessary  for  the  formation  of  the  science,  and  no  more 
than  is  necessary.  We  will,  however,  attempt  to  detect 
the  principles  which  really  must  form  the  basis  of  theo 
retical  arithmetic. 

2.  Why  is  it  that  three  and  two  are  equal  to  four  and 
one?     Because  if  we  look  at  five  things  of  any  kind,  we 
see  that  it  is  so.     The  five  are  four  and  one ;  they  are 
also  three  and  two.     The  truth   of  our  assertion  is  in 
volved  in  our  being  able  to  conceive  the  number  five  at 
all.     We  perceive  this  truth  by  intuition,  for  we  cannot 
see,  or  imagine  we  see,  five  things,  without  perceiving 
also  that  the  assertion  above  stated  is  true. 

But  how  do  we  state  in  words  this  fundamental  prin 
ciple  of  the  doctrine  of  numbers  ?  Let  us  consider  a 
very  simple  case.  If  we  wish  to  show  that  seven  and 
two  are  equal  to  four  and  five,  we  say  that  seven  are  four 
and  three,  therefore  seven  and  two  are  four  and  three 
and  two ;  and  because  three  and  two  are  five,  this  is  four 
and  five.  Mathematical  reasoners  justify  the  first  infer 
ence  (marked  by  the  conjunctive  word  therefore),  by 
saying  that  "When  equals  are  added  to  equals  the 
wholes  are  equal,"  and  that  thus,  since  seven  is  equal 
to  three  and  four,  if  we  add  two  to  both,  seven  and  two 
are  equal  to  four  and  three  and  two. 

3.  Such  axioms  as  this,  that  when  equals  are  added 
to  equals  the  wholes  are  equal,  are,  in  fact,  expressions 
of  the  general  condition  of  intuition,  by  which  a  whole 
is  contemplated  as  made  up  of  parts,  and  as  identical 
with  the  aggregate  of  the  parts.     And  a  yet  more 


ral  form  in  which  we  might  more  adequately  express 
this  conditon  of  intuition  would  be  this  ;  that  "  Two  mag 
nitudes  are  equal  when  they  can  be  divided  into  parts 
which  are  equal,  each  to  each."  Thus  in  the  above  ex 
ample,  seven  and  two  are  equal  to  four  and  five,  because 
each  of  the  two  sums  can  be  divided  into  the  parts,  four, 
three,  and  two. 

4.  In  all  these  cases,  a  person  who  had  never  seen 
such  axioms  enunciated  in  a  verbal  form  would  employ 
the  same  reasoning  as  a  practised  mathematician,  in  order 
to  satisfy  himself  that  the  proposition  was  true.     The 
steps  of  the  reasoning,  being  seen  to  be  true  by  intuition, 
would  carry  an  entire  conviction,  whether    or   not  the 
argument   were   made   verbally   complete.     Hence   the 
axioms   may  appear   superfluous,  and  on  this  account 
such  axioms  have  often  been  spoken  contemptuously  of 
as  empty  and  barren  assertions.     In  fact,   however,  al 
though  they  cannot  supply  the  deficiency  of  the  clear  in 
tuition  of  number  and  space  in  the  reasoner  himself,  and 
although  when  he  possesses  such  a  faculty,  he  will  reason 
rightly  if  he  have  never  heard  of  such  axioms,  they  still 
have  their  place  properly  at  the  beginning  of  our  trea 
tises  on  the  science  of  quantity ;  since  they  express,  as 
simply  as   words   can    express,  those  conditions  of  the 
intuition  of  magnitudes  on  which  all  reasoning  concern 
ing  quantity  must  be  based ;  and  are  necessary  when  we 
want,  not  only  to  see  the  truth  of  the  elementary  reason 
ings  on  these  subjects,  but  to  put  such  reasonings  in  a 
formal  and  logical  shape. 

5.  We  have  considered  the  above-mentioned  axioms 
as  the  basis  of  all  arithmetical  operations  of  the  nature 
of  addition.     But  it  is  easily  seen  that  the  same  prin 
ciple  may  be  carried  into  other  cases ;  as  for  instance, 
multiplication,    which    is   merely  a   repeated    addition, 
and    admits    of   the    same    kind    of    evidence.       Thus 

OF    THE    AXIOMS    WHICH    RELATE    TO    NUMIJEK.       135 

five  times  three  are  equal  to  three  times  five ;  why 
is  this  ?  If  we  arrange  fifteen  things  in  five  rows  of 
three,  it  is  seen  by  looking,  or  by  imaginary  looking, 
which  is  intuition,  that  they  may  also  be  taken  as  three 
rows  of  five.  And  thus  the  principle  that  those  wholes 
are  equal  which  can  be  resolved  into  the  same  partial 
magnitudes,  is  immediately  applicable  in  this  as  in  the 
other  case. 

C.  We  may  proceed  to  higher  numbers,  and  may  find 
ourselves  obliged  to  use  artificial  nomenclature  and 
notation  in  order  to  represent  and  reckon  them  ;  but  the 
reasoning  in  these  cases  also  is  still  the  same.  And  the 
usual  artifice  by  which  our  reasoning  in  such  instances 
is  assisted  is,  that  the  number  which  is  the  root  of  our 
scale  of  notation  (which  is  ten  in  our  usual  system),  is 
alternately  separated  into  parts  and  treated  as  a  single 
thing.  Thus  47  and  35  are  82 ;  for  47  is  four  tens  and 
seven  ;  35  is  three  tens  and  five ;  whence  47  and  35  are 
seven  tens  and  twelve ;  that  is,  7  tens,  1  ten,  and  2 ; 
which  is  8  tens  and  2,  or  82.  The  like  reasoning  is 
applicable  in  other  cases.  And  since  the  most  remote 
and  complex  properties  of  numbers  are  obtained  by  a 
prolongation  of  a  course  of  reasoning  exactly  similar  to 
that  by  which  we  thus  establish  the  most  elementary 
propositions,  we  have,  in  the  principles  just  noticed,  the 
foundation  of  the  whole  of  Theoretical  Arithmetic. 


1.  OUR  perception  of  the  passage  of  time  involves  a 
series  of  acts  of  memory.  This  is  easily  seen  and  assented 
to,  when  large  intervals  of  time  and  a  complex  train  of 
occurrences  are  concerned.  But  since  memory  is  requi- 


site  iii  order  to  apprehend  time  in  such  cases,  we  cannot 
doubt  that  the  same  faculty  must  be  concerned  in  the 
shortest  and  simplest  cases  of  succession ;  for  it  will 
hardly  be  maintained  that  the  process  by  which  we  con 
template  the  progress  of  time  is  different  when  small 
and  when  large  intervals  are  concerned.  If  memory  be 
absolutely  requisite  to  connect  two  events  which  begin 
and  end  a  day,  and  to  perceive  a  tract  of  time  between 
them,  it  must  be  equally  indispensable  to  connect  the 
beginning  and  end  of  a  minute,  or  a  second ;  though  in 
this  case  the  effort  may  be  smaller,  and  consequently 
more  easily  overlooked.  In  common  cases,  we  are  un 
conscious  of  the  act  of  thought  by  which  we  recollect 
the  preceding  instant,  though  we  perceive  the  effort  when 
we  recollect  some  distant  event.  And  this  is  analogous 
to  what  happens  in  other  instances.  Thus,  we  walk 
without  being  conscious  of  the  volitions  by  which  we 
move  our  muscles ;  but,  in  order  to  leap,  a  distinct  and 
manifest  exertion  of  the  same  muscles  is  necessary.  Yet 
no  one  will  doubt  that  we  walk  as  well  as  leap  by  an 
act  of  the  will  exerted  through  the  muscles ;  and  in  like 
manner,  our  consciousness  of  small  as  well  as  large  inter 
vals  of  time  involves  something  of  the  nature  of  an  act 
of  memory. 

2.  But  this  constant  and  almost  imperceptible  kind 
of  memory,  by  which  we  connect  the  beginning  and  end 
of  each  instant  as  it  passes,  may  very  fitly  be  distinguished 
in  common  cases  from  manifest  acts  of  recollection, 
although  it  may  be  difficult  or  impossible  to  separate 
the  two  operations  in  general.  This  perpetual  and  latent 
kind  of  memory  may  be  termed  a  sense  of  successive 
ness ;  and  must  be  considered  as  an  internal  sense  by 
which  we  perceive  ourselves  existing  in  time,  much  in 
the  same  way  as  by  our  external  and  muscular  sense 
we  perceive  ourselves  existing  in  space.  And  both  our 


internal  thoughts  and  feelings,  and  the  events  which 
take  place  around  us,  are  apprehended  as  objects  of  this 
internal  sense,  and  thus  as  taking  place  in  time. 

3.  In  the  same  manner  in  which  our  interpretation 
of  the  notices  of  the  muscular  sense  implies  the  power  of 
moving  our  limbs,  and  of  touching  at  will  this  object  or 
that ;  our  apprehension  of  the  relations  of  time  by  means 
of  the  internal  sense  of  successiveness  implies  a  power  of 
recalling  what  has  past,  and  of  retaining  what  is  pass 
ing.     We  are  able  to  seize  the  occurrences  which  have 
just  taken   place,  and  to  hold  them  fast  in  our  minds 
so  as  mentally  to  measure  their  distance  in  time  from 
occurrences  now  present.     And  thus,  this  sense  of  suc 
cessiveness,  like  the  muscular  sense  with  which  we  have 
compared  it,  implies  activity  of  the  mind  itself,   and  is 
not  a  sense  passively  receiving  impressions. 

4.  The  conception  of  Number  appears  to  require  the 
exercise  of  the  same  sense  of  succession.     At  first  sight, 
indeed,  we  seem  to  apprehend  Number  without  any  act 
of  memory,  or  any  reference  to  time :  for  example,  we 
look  at  a  horse,  and  see  that  his  legs  are  four ;  and  this 
we  seem  to  do  at  once,  without  reckoning  them.     But  it 
is  not  difficult  to  see  that  this  seeming  instantaneousness 
of  the  perception  of  small  numbers  is  an  illusion.     This 
resembles  the  many  other  cases  in  which  we  perform 
short  and  easy  acts  so  rapidly  and  familiarly  that  we  are 
unconscious  of  them;  as  in  the  acts  of  seeing,  and  of  arti 
culating  our  words.     And  this  is  the  more  manifest,  since 
we   begin  our  acquaintance  with  number  by  counting 
even  the   smallest   numbers.      Children  and  very  rude 
savages  must   use  an  effort  to  reckon   even  their  five 
fingers,  and  find  a  difficulty  in  going  further.     And  per 
sons  have  been  known  who  were  able  by  habit,  or  by  a 
peculiar  natural  aptitude,  to  count  by  dozens  as  rapidly 
as  common  persons  can  by  units.     We  may  conclude, 


therefore,  that  when  we  appear  to  catch  a  small  number 
by  a  single  glance  of  the  eye,  we  do  in  fact  count  the 
units  of  it  in  a  regular,  though  very  brief  succession.  To 
count  requires  an  act  of  memory.  Of  this  we  are  sen 
sible  when  we  count  very  slowly,  as  when  we  reckon  the 
strokes  of  a  church-clock ;  for  in  such  a  case  we  may 
forget  in  the  intervals  of  the  strokes,  and  miscount.  Now 
it  will  not  be  doubted  that  the  nature  of  the  process  in 
counting  is  the  same  whether  we  count  fast  or  slow. 
There  is  no  definite  speed  of  reckoning  at  which  the 
faculties  which  it  requires  are  changed;  and  therefore 
memory,  which  is  requisite  in  some  cases,  must  be  so 
in  all*. 

The  act  of  counting,  (one,  two,  three,  and  so  on,)  is 
the  foundation  of  all  our  knowledge  of  number.  The 
intuition  of  the  relations  of  number  involves  this  act  of 
counting;  for,  as  we  have  just  seen,  the  conception  of 
number  cannot  be  obtained  in  any  other  way.  And  thus 
the  whole  of  theoretical  arithmetic  depends  upon  an  act 
of  the  mind,  and  upon  the  conditions  which  the  exercise 
of  that  act  implies.  These  have  been  already  explained 
in  the  last  chapter. 

5.  But  if  the  apprehension  of  number  be  accompanied 
by  an  act  of  the  mind,  the  apprehension  of  rhythm  is  so 
still  more  clearly.  All  the  forms  of  versification  and  the 
measures  of  melodies  are  the  creations  of  man,  who  thus 
realizes  in  words  and  sounds  the  forms  of  recurrence 
which  rise  within  his  own  mind.  When  we  hear  in  a 

*  I  have  considered  Number  as  involving  the  exercise  of  the  sense 
of  succession,  because  I  cannot  draw  any  line  between  those  cases  of 
large  numbers,  in  which,  the  process  of  counting  being  performed,  there 
is  a  manifest  apprehension  of  succession ;  and  those  cases  of  small  num 
bers,  in  which  we  seem  to  see  the  number  at  one  glance.  But  if  any 
one  holds  Number  to  be  apprehended  by  a  direct  act  of  intuition,  as 
Space  and  Time  are,  this  view  will  not  disturb  the  other  doctrines 
delivered  .in  the  text. 


quiet  scene  any  rapidly-repeated  sound,  as  those  made  by 
the  hammer  of  the  smith  or  the  saw  of  the  carpenter, 
every  one  knows  how  insensibly  we  throw  these  noises 
into  a  rhythmical  form  in  our  own  apprehension.  We 
do  this  even  without  any  suggestion  from  the  sounds 
themselves.  For  instance,  if  the  beats  of  a  clock  or 
watch  be  ever  so  exactly  alike,  we  still  reckon  them 
alternately  tick-tack,  tick-tack.  That  this  is  the  case, 
may  be  proved  by  taking  a  watch  or  clock  of  such  a  con 
struction  that  the  returning  swing  of  the  pendulum  is 
silent,  and  in  which  therefore  all  the  beats  are  rigorously 
alike :  we  shall  find  ourselves  still  reckoning  its  sounds 
as  tick-tack.  In  this  instance  it  is  manifest  that  the 
rhythm  is  entirely  of  our  own  making.  In  melodies, 
also,  and  in  verses  in  which  the  rhythm  is  complex,  ob 
scure,  and  difficult,  we  perceive  something  is  required 
on  our  part ;  for  we  are  often  incapable  of  contributing 
our  share,  and  thus  lose  the  sense  of  the  measure  alto 
gether.  And  when  we  consider  such  cases,  and  attend 
to  what  passes  within  us  when  we  catch  the  measure, 
even  of  the  simplest  and  best-known  air,  we  shall  no 
longer  doubt  that  an  act  of  our  own  thoughts  is  requisite 
in  such  cases,  as  well  as  impressions  on  the  sense.  And 
thus  the  conception  of  this  peculiar  modification  of  time, 
which  we  have  called  rhythm,  like  all  the  other  views 
which  we  have  taken  of  the  subject,  shows  that  we  must, 
in  order  to  form  such  conceptions,  supply  a  certain  idea 
by  our  own  thoughts,  as  well  as  merely  receive  by  senses, 
whether  external  or  internal,  the  impressions  of  appear 
ances  and  collections  of  appearances. 


I  HAVE  in  the  last  ten  chapters  described  Space,  Time,  and  Xunilwr  by 
various  expressions,  all  intended  to  point  out  their  office  as  exemplifying 
the  Ideal  Element  of  human  knowledge.  I  have  called  them  Funda- 


mental  Ideas  ;  Forms  of  Perception  ;  Forms  of  Intuition  ;  and  per 
haps  other  names.  I  might  add  yet  other  phrases.  I  might  say  that 
the  properties  of  Space,  Time,  and  Number  are  Laws  of  the  Mind's 
Activity  in  apprehending  what  is.  For  the  mind  cannot  apprehend  any 
thing  or  event  except  conformably  to  the  properties  of  space,  time,  and 
number.  It  is  not  only  that  it  does  not,  but  it  can  not  :  and  this 
impossibility  shows  that  the  law  is  a  law  of  the  mind,  and  not  of 
objects  extraneous  to  the  mind. 

It  is  usual  for  some  of  those  who  reject  the  doctrines  here  presented 
to  say  that  the  axioms  of  geometry,  and  of  other  sciences,  are  obtained 
by  Induction  from  facts  constantly  presented  by  experience.  But  I  do 
not  see  how  Induction  can  prove  that  a  proposition  must  be  true.  The 
only  intelligible  usage  of  the  word  Induction  appears  to  me  to  be,  that  in 
which  it  is  applied  to  a  proposition  which,  being  separable  from  the 
facts  in  our  apprehension,  and  being  compared  with  them,  is  seen  to 
agree  with  them.  But  in  the  cases  now  spoken  of,  the  proposition  is 
not  separable  from  the  facts.  We  cannot  infer  by  induction  that  two 
straight  lines  cannot  inclose  a  space,  because  we  cannot  contemplate 
special  cases  of  two  lines  inclosing  a  space,  in  which  it  remains  to  be 
determined  whether  or  not  the  proposition,  that  both  are  straight, 
is  true. 

I  do  not  deny  that  the  activity  of  the  mind  by  which  it  perceives 
objects  and  events  as  related  according  to  the  laws  of  space,  time,  and 
number,  is  awakened  and  developed  by  being  constantly  exercised ;  and 
that  we  cannot  imagine  a  stage  of  human  existence  in  which  the  powers 
have  not  been  awakened  and  developed  by  such  exercise.  In  this  way, 
experience  and  observation  are  necessary  conditions  and  prerequisites  of 
our  apprehension  of  geometrical  (and  other)  axioms.  We  cannot  see 
the  truth  of  these  axioms  without  some  experience,  because  we  cannot 
see  any  thing,  or  be  human  beings,  without  some  experience.  This 
might  be  expressed  by  saying  that  such  truths  are  acquired  necessarily 
in  the  course  of  all  experience ;  but  I  think  it  is  very  undesirable  to 
apply,  to  such  a  case,  the  word  Induction,  of  which  it  is  so  important 
to  us  to  keep  the  scientific  meaning  free  from  confusion.  Induction 
cannot  give  demonstrative  proofs,  as  I  have  already  stated  in  Book  i. 
C.  ii.  sect.  3,  and  therefore  cannot  be  the  ground  of  necessary  truths. 

Another  expression  which  may  be  used  to  describe  the  Funda 
mental  Ideas  here  spoken  of  is  suggested  by  the  language  of  a  very 
profound  and  acute  Review  of  the  former  edition.  The  Reviewer  holds 
that  we  pass  from  special  experiences  to  universal  truths  in  virtue  of 
*'  the  inductive  propensity — the  irresistible  impulse  of  the  mind  to 
generalize  ad  in/initnm."  I  have  already  given  reasons  why  I  cannot 
adopt  the  former  expression  ;  but  I  do  not  see  why  space,  time,  number, 


cause,  and  the  rest,  may  not  IK?  termed  different  forms  of  the  impulse  of 
the  mind  to  generalize.  If  we  put  together  all  the  Fundamental  Ideas 
as  results  of  the  Generalizing  Impulse,  we  must  still  separate  them  as 
different  modes  of  action  of  that  Impulse,  showing  themselves  in  various 
characteristic  ways  in  the  axioms  and  modes  of  reasoning  which  belong 
to  different  sciences.  The  Generalizing  Impulse  in  one  case  proceeds 
according  to  the  Idea  of  Space  ;  in  another,  according  to  the  Idea  of 
Mechanical  Cause ;  and  so  in  other  subjects. 


1.  Discursive  Reasoning. — WE  have  thus  seen  that 
our  notions  of  space,  time,  and  their  modifications,  neces 
sarily  involve  a  certain  activity  of  the  mind ;  and  that 
the  conditions  of  this  activity  form  the  foundations  of 
those  sciences  which  have  the  relations  of  space,  time, 
and  number,  for  their  object.  Upon  the  fundamental 
principles  thus  established,  the  various  sciences  which 
are  included  in  the  term  Pure  Mathematics,  (Geometry, 
Algebra,  Trigonometry,  Conic  Sections,  and  the  rest  of 
the  Higher  Geometry,  the  Differential  Calculus,  and  the 
like,)  are  built  up  by  a  series  of  reasonings.  These  rea 
sonings  are  subject  to  the  rules  of  Logic,  as  we  have 
already  remarked ;  nor  is  it  necessary  here  to  dwell  long 
on  the  nature  and  rules  of  such  processes.  But  we  may 
here  notice  that  such  processes  are  termed  discursire, 
in  opposition  to  the  operations  by  which  we  acquire  our 
fundamental  principles,  which  are,  as  we  have  seen,  intui 
tive.  This  opposition  was  formerly  very  familiar  to  our 
writers  ;  as  Milton,— 

.     .     .     Thus  the  soul  reason  receives, 
Discursive  or  intuitive. — Paradise  Lost,  \.  438. 

For  in  such  reasonings  we  obtain  our  conclusions,  not 
by  looking  at  our  conceptions  steadily  in  one  view,  which 


is  intuition,  but  by  passing  from  one  view  to  another,  like 
those  who  run  from  place  to  place  (discursus).  Thus  a 
straight  line  may  be  at  the  same  time  a  side  of  a  triangle 
and  a  radius  of  a  circle :  and  in  the  first  proposition  of 
Euclid  a  line  is  considered,  first  in  one  of  these  relations, 
and  then  in  the  other,  and  thus  the  sides  of  a  certain 
triangle  are  proved  to  be  equal.  And  by  this  "  discourse 
of  reason,"  as  by  our  older  writers  it  was  termed,  we  set 
forth  from  those  axioms  which  we  perceive  by  intuition, 
travel  securely  over  a  vast  and  varied  region,  and  become 
possessed  of  a  copious  store  of  mathematical  truths. 

2.  Technical  Terms  of  Reasoning. — The  reasoning  of 
mathematics,  thus  proceeding  from  a  few  simple  princi 
ples  to  many  truths,  is  conducted  according  to  the  rules 
of  Logic.  If  it  be  necessary,  mathematical  proofs  may  be 
reduced  to  logical  forms,  and  expressed  in  Syllogisms, 
consisting  of  major,  minor,  and  conclusion.  But  in  most 
cases  the  syllogism  is  of  that  kind  which  is  called  by 
logical  writers  an  Enthymeme;  a  word  which  implies 
something  existing  in  the  thoughts  only,  and  which  desig 
nates  a  syllogism  in  which  one  of  the  premises  is  under 
stood,  and  not  expressed.  Thus  we  say  in  a  mathematical 
proof,  "  because  the  point  c  is  the  center  of  the  circle  AB, 
AC  is  equal  to  BC  ;"  not  stating  the  major, — that  all  lines 
drawn  from  the  center  of  a  circle  to  the  circumference 
are  equal;  or  introducing  it  only  by  a  transient  reference 
to  the  definition  of  a  circle.  But  the  enthymeme  is  so 
constantly  used  in  all  habitual  forms  of  reasoning,  that 
it  does  not  occur  to  us  as  being  anything  peculiar  in 
mathematical  works. 

The  propositions  which  are  proved  to  be  generally 
true  are  termed  Theorems:  but  when  anything  is  required 
to  be  done,  as  to  draw  a  line  or  a  circle  under  given 
conditions,  this  proposition  is  a  Problem.  A  theorem  re 
quires  demonstration;  a  problem,  solution.  And  for  both 


purposes  the  mathematician  usually  makes  a  Constrw- 
tion.  He  directs  us  to  draw  certain  lines,  circles,  or  other 
curves,  on  which  is  to  be  founded  his  demonstration  that 
his  theorem  is  true,  or  that  his  problem  is  solved.  Some 
times,  too,  he  establishes  some  Lemma,  or  preparatory 
proposition,  before  he  proceeds  to  his  main  task ;  and 
often  he  deduces  from  his  demonstration  some  conclusion 
in  addition  to  that  which  was  the  professed  object  of  his 
proposition ;  and  this  is  termed  a  Corollary. 

These  technical  terms  are  noted  here,  not  as  being 
very  important,  but  in  order  that  they  may  not  sound 
strange  and  unintelligible  if  we  should  have  occasion  to 
use  some  of  them.  There  is,  however,  one  technical  dis 
tinction  more  peculiar,  and  more  important. 

3.  Geometrical  Analysis  and  Synthesis. — In  geome 
trical  reasoning  such  as  we  have  described,  we  introduce 
at  every  step  some  new  consideration ;  and  it  is  by  com 
bining  all  these  considerations,  that  we  arrive  at  the 
conclusion,  that  is,  the  demonstration  of  the  proposition. 
Each  step  tends  to  the  final  result,  by  exhibiting  some 
part  of  the  figure  under  a  new  relation.  To  what  we 
have  already  proved,  is  added  something  more;  and  hence 
this  process  is  called  Synthesis,  or  putting  together.  The 
proof  flows  on,  receiving  at  every  turn  new  contribu 
tions  from  different  quarters ;  like  a  river  fed  and  aug 
mented  by  many  tributary  streams.  And  each  of  these 
tributaries  flows  from  some  definition  or  axiom  as  its 
fountain,  or  is  itself  formed  by  the  union  of  smaller  rivulets 
which  have  sources  of  this  kind.  In  descending  along  its 
course,  the  synthetical  proof  gathers  all  these  accessions 
into  one  common  trunk,  the  proposition  finally  proved. 

But  we  may  proceed  in  a  different  manner.      We 

mav   besin   from   the   formed  river,   and  ascend  to   its 


sources.  We  may  take  the  proposition  of  which  we 
require  a  proof,  and  may  examine  what  the  supposition 


of  its  truth  implies.  If  this  be  true,  then  something  else 
may  be  seen  to  be  true ;  and  from  this,  something  else, 
and  so  on.  We  may  often,  in  this  way,  discover  of  what 
simpler  propositions  our  theorem  or  solution  is  com 
pounded,  and  may  resolve  these  in  succession,  till  we 
come  to  some  proposition  which  is  obvious.  This  is  geo 
metrical  Analysis.  Having  succeeded  in  this  analytical 
process,  we  may  invert  it ;  and  may  descend  again  from 
the  simple  and  known  propositions,  to  the  proof  of  a 
theorem,  or  the  solution  of  a  problem,  which  was  our 

This  process  resembles,  as  we  have  said,  tracing  a 
river  to  its  sources.  As  we  ascend  the  stream,  we  per 
petually  meet  with  bifurcations ;  and  some  sagacity  is 
needed  to  enable  us  to  see  which,  in  each  case,  is  the 
main  stream :  but  if  we  proceed  in  our  'research,  we 
exhaust  the  unexplored  valleys,  and  finally  obtain  a  clear 
knowledge  of  the  place  whence  the  waters  flow.  Analy 
tical  is  sometimes  confounded  with  symbolical  reasoning, 
on  which  subject  we  shall  make  a  remark  in  the  next 
chapter.  The  object  of  that  chapter  is  to  notice  certain 
other  fundamental  principles  and  ideas,  not  included  in 
those  hitherto  spoken  of,  which  we  find  thrown  in  our 
way  as  we  proceed  in  our  mathematical  speculations. 
It  would  detain  us  too  long,  and  involve  us  in  subtle  and 
technical  disquisitions,  to  examine  fully  the  grounds  of 
these  principles ;  but  the  Mathematics  hold  so  important 
a  place  in  relation  to  the  inductive  sciences,  that  I  shall 
briefly  notice  the  leading  ideas  which  the  ulterior  pro 
gress  of  the  subject  involves. 




I.  The  Idea  of  a  Limit. — THE  general  truths  concern 
ing  relations  of  space  which  depend  upon  the  axioms 
and  definitions  contained  in  Euclid's  Elements,  and  which 
involve  only  properties  of  straight  lines  and  circles,  are 
termed  Elementary  Geometry :  all  beyond  this  belongs  to 
the  Higher  Geometry.  To  this  latter  province  appertain, 
for  example,  all  propositions  respecting  the  lengths  of  any 
portions  of  curve  lines ;  for  these  cannot  be  obtained  by 
means  of  the  principles  of  the  Elements  alone.  Here 
then  we  mus£  ask  to  what  other  principles  the  geometer 
has  recourse,  and  from  what  source  these  are  drawn.  Is 
there  any  origin  of  geometrical  truth  which  we  have  not 
yet  explored  ? 

The  Idea  of  a  Limit  supplies  a  new  mode  of  establish 
ing  mathematical  truths.  Thus  with  regard  to  the  length 
of  any  portion  of  a  curve,  a  problem  which  we  have  just 
mentioned ;  a  curve  is  not  made  up  of  straight  lines,  and 
therefore  we  cannot  by  means  of  any  of  the  doctrines  of 
elementary  geometry  measure  the  length  of  any  curve. 
But  we  may  make  up  a  figure  nearly  resembling  any 
curve  by  putting  together  many  short  straight  lines,  just 
as  a  polygonal  building  of  very  many  sides  may  nearly 
resemble  a  circular  room.  And  in  order  to  approach 
nearer  and  nearer  to  the  curve,  we  may  make  the  sides 
more  and  more  small,  more  and  more  numerous.  We 
may  then  possibly  find  some  mode  of  measurement,  some 
relation  of  these  small  lines  to  other  lines,  which  is  not 
disturbed  by  the  multiplication  of  the  sides,  however  far 
it  be  carried.  And  thus,  we  may  do  what  is  equivalent  to 

VOL.  i.    \v.  P.  L 


measuring  the  curve  itself;  for  by  multiplying  the  sides 
we  may  approach  more  and  more  closely  to  the  curve  till 
no  appreciable  difference  remains.  The  curve  line  is  the 
Limit  of  the  polygon ;  and  in  this  process  we  proceed  on 
the  Axiom,  that  "What  is  true  up  to  the  limit  is  true  at 
the  limit." 

This  mode  of  conceiving  mathematical  magnitudes  is 
of  wide  extent  and  use ;  for  every  curve  may  be  con 
sidered  as  the  limit  of  some  polygon ;  every  varied 
magnitude,  as  the  limit  of  some  aggregate  of  simpler 
forms ;  and  thus  the  relations  of  the  elementary  figures 
enable  us  to  advance  to  the  properties  of  the  most  com 
plex  cases. 

A  Limit  is  a  peculiar  and  fundamental  conception,  the 
use  of  which  in  proving  the  propositions  of  the  Higher 
Geometry  cannot  be  superseded  by  any  combination  of 
other  hypotheses  and  definitions*.  The  axiom  just  no 
ticed,  that  what  is  true  up  to  the  limit  is  true  at  the  limit, 
is  involved  in  the  very  conception  of  a  limit :  and  this 
principle,  with  its  consequences,  leads  to  all  the  results 
which  form  the  subject  of  the  higher  mathematics,  whe- 

*  This  assertion  cannot  be  fully  proved  and  illustrated  without  a 
reference  to  mathematical  reasonings  which  would  not  be  generally 
intelligible.  I  have  shown  the  truth  of  the  assertion  in  my  Thoughts 
on  the  Study  of  Mathematics,  annexed  to  the  Principles  of  English 
University  Education.  The  proof  is  of  this  kind : — The  ultimate 
equality  of  an  arc  of  a  curve  and  the  corresponding  periphery  of  a 
polygon,  when  the  sides  of  the  polygon  are  indefinitely  increased  in 
number,  is  evident.  But  this  truth  cannot  be  proved  from  any  other 
axiom.  For  if  we  take  the  supposed  axiom,  that  a  curve  is  always 
less  than  the  including  broken  line,  this  is  not  true,  except  with  a  con 
dition  ;  and  in  tracing  the  import  of  this  condition,  we  find  its  neces 
sity  becomes  evident  only  when  we  introduce  a  reference  to  a  Limit. 
And  the  same  is  the  case  if  we  attempt  to  supersede  the  notion  of  a 
Limit  in  proving  any  other  simple  and  evident  proposition  in  which 
that  notion  is  involved.  Therefore  these  evident  truths  are  self-evident, 
in  virtue  of  /lie  Idea  of  a  Limit. 


thcr  proved  by  the  consideration  of  evanescent  triangles, 
by  the  processes  of  the  Differential  Calculus,  or  in  any 
other  way. 

The  ancients  did  not  expressly  introduce  this  con 
ception  of  a  Limit  into  their  mathematical  reasonings ; 
although  in  the  application  of  what  is  termed  the 
Method  of  Exhaustions,  (in  which  they  show  how  to 
exhaust  the  difference  between  a  polygon  and  a  curve,  or 
the  like,)  they  were  in  fact  proceeding  upon  an  obscure 
apprehension  of  principles  equivalent  to  those  of  the 
Method  of  Limits.  Yet  the  necessary  fundamental  prin 
ciple  not  having,  in  their  time,  been  clearly  developed, 
their  reasonings  were  both  needlessly  intricate  and  im 
perfectly  satisfactory.  Moreover  they  were  led  to  put  in 
the  place  of  axioms,  assumptions  which  were  by  no  means 
self-evident ;  as  when  Archimedes  assumed,  for  the  basis 
of  his  measure  of  the  circumference  of  the  circle,  the 
proposition  that  a  circular  arch  is  necessarily  less  than 
two  lines  which  inclose  it,  joining  its  extremities.  The 
reasonings  of  the  older  mathematicians,  which  professed 
to  proceed  upon  such  assumptions,  led  to  true  results 
in  reality,  only  because  they  were  guided  by  a  latent 
reference  to  the  limiting  case  of  such  assumptions.  And 
this  latent  employment  of  the  conception  of  a  Limit, 
reappeared  in  various  forms  during  the  early  period  of 
modern  mathematics ;  as  for  example,  in  the  Method  of 
Indivisibles  of  Cavalleri,  and  the  Characteristic  Triangle 
of  Barrow ;  till  at  last,  Newton  distinctly  referred  such 
reasonings  to  the  conception  of  a  Limit,  and  established 
the  fundamental  principles  and  processes  which  that 
conception  introduces,  with  a  distinctness  and  exactness 
which  required  little  improvement  to  make  it  as  unim 
peachable  as  the  demonstrations  of  geometry.  And  when 
such  processes  as  Newton  thus  deduced  from  the  con 
ception  of  a  Limit  are  represented  by  means  of  general 



algebraical  symbols  instead  of  geometrical  diagrams,  we 
have  then  before  us  the  Method  of  Fluxions,  or  the 
Differential  Calculus;  a  mode  of  treating  mathematical 
problems  justly  considered  as  the  principal  weapon  by 
which  the  splendid  triumphs  of  modern  mathematics 
have  been  achieved. 

2.  The  Use  of  General  Symbols. — The  employment 
of  algebraical  symbols,  of  which  we  have  just  spoken, 
has  been  another  of  the  main  instruments  to  which  the 
successes  of  modern  mathematics  are  owing.  And  here 
again  the  processes  by  which  we  obtain  our  results  de 
pend  for  their  evidence  upon  a  fundamental  conception, 
— the  conception  of  arbitrary  symbols  as  the  Signs  of 
quantity  and  its  relations ;  and  upon  a  corresponding 
axiom,  that  "  The  interpretation  of  such  symbols  must 
be  perfectly  general."  In  this  case,  as  in  the  last,  it  was 
only  by  degrees  that  mathematicians  were  led  to  a  just 
apprehension  of  the  grounds  of  their  reasoning.  For 
symbols  were  at  first  used  only  to  represent  numbers 
considered  with  regard  to  their  numerical  properties ; 
and  thus  the  science  of  Algebra  was  formed.  But  it  was 
found,  even  in  cases  belonging  to  common  algebra,  that 
the  symbols  often  admitted  of  an  interpretation  which 
went  beyond  the  limits  of  the  problem,  and  which  yet  was 
not  unmeaning,  since  it  pointed  out  a  question  closely 
analogous  to  the  question  proposed.  This  was  the  case, 
for  example,  when  the  answer  was  a  negative  quantity ; 
for  when  Descartes  had  introduced  the  mode  of  repre 
senting  curves  by  means  of  algebraical  relations  among 
the  symbols  of  the  co-ordinates,  or  distances  of  each  of 
their  points  from  fixed  lines,  it  was  found  that  negative 
quantities  must  be  dealt  with  as  not  less  truly  significant 
than  positive  ones.  And  as  the  researches  of  mathema 
ticians  proceeded,  other  cases  also  were  found,  in  which 
the  symbols,  although  destitute  of  meaning  according  to 


the  original  conventions  of  their  institution,  still  pointed 
out  truths  which  could  be  verified  in  other  ways ;  as  in 
the  cases  in  which  what  are  called  Impossible  quantities 
occur.  Such  processes  may  usually  be  confirmed  upon 
other  principles,  and  the  truth  in  question  may  be  esta 
blished  by  means  of  a  demonstration  in  which  no  such 
secerning  fallacies  defeat  the  reasoning.  But  it  has  also 
been  shown  in  many  such  cases,  that  the  process  in  which 
some  of  the  steps  appear  to  be  without  real  meaning, 
does  in  fact  involve  a  valid  proof  of  the  proposition. 
And  what  we  have  here  to  remark  is,  that  this  is  not 
true  accidentally  or  partially  only,  but  that  the  results 
of  systematic  symbolical  reasoning  must  always  express 
general  truths,  by  their  nature,  and  do  not,  for  their 
justification,  require  each  of  the  steps  of  the  process  to 
represent  some  definite  operation  upon  quantity.  The 
absolute  universality  of  the  interpretation  of  symbols  is 
the  fundamental  principle  of  their  use.  This  has  been 
shown  very  ably  by  Dr.  Peacock  in  his  Alyebra.  He 
has  there  illustrated,  in  a  variety  of  ways,  this  prin 
ciple  :  that  "  If  general  symbols  express  an  identity 
when  they  are  supposed  to  be  of  any  special  nature, 
they  must  also  express  an  identity  when  they  are  gene 
ral  in  their  nature."  And  thus,  this  universality  of  sym 
bols  is  a  principle  in  addition  to  those  we  have  already 
noticed ;  and  is  a  principle  of  the  greatest  importance 
in  the  formation  of  mathematical  science,  according  to 
the  wide  generality  which  such  science  has  in  modern 
times  assumed. 

3.  Connexion  of  Symbols  and  Analysis. — Since  in 
our  symbolical  reasoning  our  symbols  thus  reason  for  us, 
we  do  not  necessarily  here,  as  in  geometrical  reasoning, 
go  on  adding  carefully  one  known  truth  to  another,  till 
we  reach  the  desired  result.  On  the  contrary,  if  we  have 
a  theorem  to  prove  or  a  problem  to  solve  which  can  be 


brought  under  the  domain  of  our  symbols,  we  may  at 
once  state  the  given  but  unproved  truth,  or  the  given 
combination  of  unknown  quantities,  in  its  symbolical 
form.  After  this  first  process,  we  may  then  proceed  to 
trace,  by  means  of  our  symbols,  what  other  truth  is 
involved  in  the  one  thus  stated,  or  what  the  unknown 
symbols  must  signify;  resolving  step  by  step  the  sym 
bolical  assertion  with  which  we  began,  into  others  more 
fitted  for  our  purpose.  The  former  process  is  a  kind  of 
synthesis,  the  latter  is  termed  analysis.  And  although 
symbolical  reasoning  does  not  necessarily  imply  such 
analysis;  yet  the  connexion  is  so  familiar,  that  the 
term  analysis  is  frequently  used  to  designate  symbolical 


1.  Pure  Mechanism,. — THE  doctrine  of  Motion,  of 
which  we  have  here  to  speak,  is  that  in  which  motion  is 
considered  quite  independently  of  its  cause,  force;  for 
all  consideration  of  force  belongs  to  a  class  of  ideas 
entirely  different  from  those  with  which  we  are  here 
concerned.  In  this  view  it  may  be  termed  the  pure 
doctrine  of  motion,  since  it  has  to  do  solely  with  space 
and  time,  which  are  the  subjects  of  pure  mathematics. 
(See  C.  I.  of  this  Book.)  Although  the  doctrine  of 
motion  in  connexion  with  force,  which  is  the  subject 
of  mechanics,  is  by  far  the  most  important  form  in 
which  the  consideration  of  motion  enters  into  the  form 
ation  of  our  sciences,  the  Pure  Doctrine  of  Motion, 
which  treats  of  space,  time,  and  velocity,  might  be  fol 
lowed  out  so  as  to  give  rise  to  a  very  considerable  and 
curious  body  of  science.  Such  a  science  is  the  science 


of  Mechanism,  independent  of  force,  and  considered  as 
the  solution  of  a  problem  which  may  be  thus  enunciated: 
*'  To  communicate  any  given  motion  from  a  first  mover 
to  a  given  body."  The  science  which  should  have  for  its 
object  to  solve  all  the  various  cases  into  which  this  pro 
blem  would  ramify,  might  be  termed  Pure  Mechanism, 
in  contradistinction  to  Mechanics  Proper,  or  Machinery, 
in  which  Force  is  taken  into  consideration.  The  greater 
part  of  the  machines  which  have  been  constructed  for 
use  in  manufactures  have  been  practical  solutions  of  some 
of  the  cases  of  this  problem.  We  have  also  important 
contributions  to  such  a  science  in  the  works  of  mathe 
maticians  ;  for  example,  the  various  investigations  and 
demonstrations  which  have  been  published  respecting 
the  form  of  the  Teeth  of  Wheels,  and  Mr.  Babbage's 
memoir*  on  the  Language  of  Machinery.  There  are 
also  several  works  which  contain  collections  of  the 
mechanical  contrivances  which  have  been  invented  for 
the  purpose  of  transmitting  and  modifying  motion,  and 
these  works  may  be  considered  as  treatises  on  the  science 
of  Pure  Mechanism.  But  this  science  has  not  yet  been 
reduced  to  the  systematic  simplicity  wfhich  is  desirable, 
nor  indeed  generally  recognized  as  a  separate  science.  It 
has  been  confounded,  under  the  common  name  of  Me 
chanics,  with  the  other  science,  Mechanics  Proper,  or 
Machinery,  which  considers  the  effect  of  force  transmitted 
by  mechanism  from  one  part  of  a  material  combination 
to  another.  For  example,  the  Mechanical  Powers,  as 
they  are  usually  termed,  (the  Lever,  the  Wheel  and 
Axle,  the  Inclined  Plane,  the  Wedge,  and  the  Screw,) 
have  almost  always  been  treated  with  reference  to  the 
relation  between  the  Power  and  the  Weight,  and  not 
primarily  as  a  mode  of  changing  the  velocity  and  kind 

*  On  a  Method  of  expressing  by  Signs  l/ic  Action  of  Machinery. 
Phil.  Trans.,  1826,  p.  250. 


of  the  motion.  The  science  of  pure  motion  has  not 
generally  been  separated  from  the  science  of  motion 
viewed  with  reference  to  its  causes. 

Recently,  indeed,  the  necessity  of  such  a  separation 
has  been  seen  by  those  who  have  taken  a  philosophical 
view  of  science.  Thus  this  necessity  has  been  urged  by 
M.  Ampere,  in  his  Essai  sur  la  Philosophic,  des  Sciences 
(1834):  "Long,"  he  says,  (p.  50),  "before  I  employed 
myself  upon  the  present  work,  I  had  remarked  that  it  is 
usual  to  omit,  in  the  brginning  of  all  books  treating  of 
sciences  which  regard  motion  and  force,  certain  consi 
derations  which,  duly  developed,  must  constitute  a  special 
science :  of  which  science  certain  parts  have  been  treated 
of,  either  in  memoirs  or  in  special  works ;  such,  for  ex 
ample,  as  that  of  Carnot  upon  Motion  considered  geome 
trically,  and  the  essay  of  Lanz  and  Betancourt  upon  the 
Composition  of  Machines."  He  then  proceeds  to  describe 
this  science  nearly  as  we  have  done,  and  proposes  to 
term  it  Kinematics  (Cinematique],  from  /ai^a,  motion. 

2.  Formal  Astronomy. — I  shall  not  attempt  here 
further  to  develop  the  form  which  such  a  science  must 
assume.  But  I  may  notice  one  very  large  province  which 
belongs  to  it.  When  men  had  ascertained  the  apparent 
motions  of  the  sun,  moon,  and  stars,  to  a  moderate 
degree  of  regularity  and  accuracy,  they  tried  to  conceive 
in  their  minds  some  mechanism  by  which  these  motions 
might  be  produced ;  and  thus  they  in  fact  proposed  to 
themselves  a  very  extensive  problem  in  Kinematics. 
This,  indeed,  was  the  view  originally  entertained  of  the 
nature  of  the  science  of  astronomy.  Thus  Plato  in  the 
seventh  Book  of  his  Republic*,  speaks  of  astronomy  as 
the  doctrine  of  the  motion  of  solids,  meaning  thereby, 
spheres.  And  the  same  was  a  proper  description  of  the 
science  till  the  time  of  Kepler,  and  even  later:  for 

4  P.  528. 


Kepler  endeavoured  in  vain  to  conjoin  with  the  know 
ledge  of  the  motions  of  the  heavenly  bodies,  those  true 
mechanical  conceptions  which  converted  formal  into 
physical  astronomy"". 

The  astronomy  of  the  ancients  admitted  none  but 
uniform  circular  motions,  and  could  therefore  be  com 
pletely  cultivated  by  the  aid  of  their  elementary  geo 
metry.  But  the  pure  science  of  motion  might  be 
extended  to  all  motions,  however  varied  as  to  the  speed 
or  the  path  of  the  moving  body.  In  this  form  it  must 
depend  upon  the  doctrine  of  limits ;  and  the  funda 
mental  principle  of  its  reasonings  would  be  this :  That 
velocity  is  measured  by  the  Limit  of  the  space  described, 
considered  with  reference  to  the  time  in  which  it  is 
described.  I  shall  not  further  pursue  this  subject ;  and 
in  order  to  complete  what  I  have  to  say  respecting  the 
Pure  Sciences,  I  have  only  a  few  words  to  add  respect 
ing  their  bearing  on  Inductive  Science  in  general. 



I.  ALL  objects  in  the  world  which  can  be  made  the 
subjects  of  our  contemplation  are  subordinate  to  the 
conditions  of  Space,  Time,  and  Number;  and  on  this 
account,  the  doctrines  of  pure  mathematics  have  most 
numerous  and  extensive  applications  in  every  depart 
ment  of  our  investigations  of  nature.  And  there  is  a 
peculiarity  in  these  Ideas,  which  has  caused  the  mathe 
matical  sciences  to  be,  in  all  cases,  the  first  successful 
efforts  of  the  awakening  speculative  powers  of  nations  at 
*  Hist.  Ind.  ,SY..  ii.  130. 


the  commencement  of  their  intellectual  progress.  Con 
ceptions  derived  from  these  Ideas  are,  from  the  very 
first,  perfectly  precise  and  clear,  so  as  to  be  fit  elements 
of  scientific  truths.  This  is  not  the  case  with  the  other 
conceptions  which  form  the  subjects  of  scientific  in 
quiries.  The  conception  of  statical  force,  for  instance, 
was  never  presented  in  a  distinct  form  till  the  works  of 
Archimedes  appeared  :  the  conception  of  accelerating 
force  was  confused,  in  the  mind  of  Kepler  and  his  con 
temporaries,  and  only  became  clear  enough  for  purposes 
of  sound  scientific  reasoning  in  the  succeeding  century : 
the  just  conception  of  chemical  composition  of  elements 
gradually,  in  modern  times,  emerged  from  the  erroneous 
and  vague  notions  of  the  ancients.  If  we  take  works 
published  on  such  subjects  before  the  epoch  when  the 
foundations  of  the  true  science  were  laid,  we  find  the 
knowledge  not  only  small,  but  worthless.  The  writers 
did  not  see  any  evidence  in  what  we  now  consider  as  the 
axioms  of  the  science ;  nor  any  inconsistency  where  we 
now  see  self-contradiction.  But  this  was  never  the  case 
with  speculations  concerning  space  and  number.  From 
their  first  rise,  these  were  true  as  far  as  they  went. 
The  Geometry  and  Arithmetic  of  the  Greeks  and  Indians, 
even  in  their  first  and  most  scanty  form,  contained  none 
but  true  propositions.  Men's  intuitions  upon  these  sub 
jects  never  allowed  them  to  slide  into  error  and  confu 
sion  ;  and  the  truths  to  which  they  were  led  by  the  first 
efforts  of  their  faculties,  so  employed,  form  part  of  the 
present  stock  of  our  mathematical  knowledge. 

2,  But  we  are  here  not  so  much  concerned  with 
mathematics  in  their  pure  form,  as  with  their  applica 
tion  to  the  phenomena  and  laws  of  nature.  And  here 
also  the  very  earliest  history  of  civilization  presents  to 
us  some  of  the  most  remarkable  examples  of  man's  suc 
cess  in  his  attempts  to  attain  to  science.  Space  and 


time,  position  and  motion,  govern  all  visible  objects ; 
but  by  far  the  most  conspicuous  examples  of  the  rela 
tions  which  arise  out  of  such  elements,  are  displayed  by 
the  ever-moving  luminaries  of  the  sky,  which  measure 
days,  and  months,  and  years,  by  their  motions,  and 
man's  place  on  the  earth  by  their  position.  Hence  the 
sciences  of  space  and  number  were  from  the  first  culti 
vated  with  peculiar  reference  to  Astronomy.  I  have 
elsewhere  *  quoted  Plato's  remark, — that  it  is  absurd 
to  call  the  science  of  the  relations  of  space  geometry, 
the  measure  of  the  earth,  since  its  most  important  office 
is  to  be  found  in  its  application  to  the  heavens.  And 
on  other  occasions  also  it  appears  how  strongly  he,  who 
may  be  considered  as  the  representative  of  the  scientific 
and  speculative  tendencies  of  his  time  and  country,  had 
been  impressed  with  the  conviction,  that  the  formation 
of  a  science  of  the  celestial  motions  must  depend  entirely 
upon  the  progress  of  mathematics.  In  the  Epilogue  to 
the  Dialogue  on  the  Laws\,  he  declares  mathematical 
knowledge  to  be  the  first  and  main  requisite  for  the 
astronomer,  and  describes  the  portions  of  it  which  he 
holds  necessary  for  astronomical  speculators  to  culti 
vate.  These  seem  to  be,  Plane  Geometry,  Theoretical 
Arithmetic,  the  Application  of  Arithmetic  to  planes 
and  to  solids,  and  finally  the  doctrine  of  Harmonics. 
Indeed  the  bias  of  Plato  appears  to  be  rather  to  con 
sider  mathematics  as  the  essence  of  the  science  of 
astronomy,  than  as  its  instrument ;  and  he  seems  dis 
posed,  in  this  as  in  other  things,  to  disparage  observa 
tion,  and  to  aspire  after  a  science  founded  upon  demon 
stration  alone.  "  An  astronomer,"  he  says  in  the  same 
place,  "must  not  be  like  Hesiod  and  persons  of  that 
kind,  whose  astronomy  consists  in  noting  the  settings 
and  risings  of  the  stars ;  but  he  must  be  one  who 

*   ///*/.  Ind.  Sc..  D.  in.  c.  ii.  t   Epinomis.  p.  990. 


understands  the  revolutions  of  the  celestial  spheres,  each 
performing  its  proper  cycle." 

A  large  portion  of  the  mathematics  of  the  Greeks, 
so  long  as  their  scientific  activity  continued,  was  directed 
towards  astronomy.  Besides  many  curious  propositions 
of  plane  and  solid  Geometry,  to  which  their  astronomers 
were  led,  their  Arithmetic,  though  very  inconvenient  in 
its  fundamental  assumptions,  was  cultivated  to  a  great 
extent ;  and  the  science  of  Trigonometry,  in  which  pro 
blems  concerning  the  relations  of  space  were  resolved  by 
means  of  tables  of  numerical  results  previously  obtained, 
was  created.  Menelaus  of  Alexandria  wrote  six  Books 
on  Chords,  probably  containing  methods  of  calculating 
Tables  of  these  quantities ;  such  Tables  were  familiarly 
used  by  the  later  Greek  astronomers.  The  same  author 
also  wrote  three  Books  on  Spherical  Trigonometry, 
which  are  still  extant. 

3.  The  Greeks,  however,  in  the  first  vigour  of  their 
pursuit  of  mathematical  truth,  at  the  time  of  Plato  and 
soon  after,  had  by  no  means  confined  themselves  to 
those  propositions  which  had  a  visible  bearing  on  the 
phenomena  of  nature  ;  but  had  followed  out  many  beau 
tiful  trains  of  research,  concerning  various  kinds  of 
figures,  for  the  sake  of  their  beauty  alone ;  as  for  in 
stance  in  their  doctrine  of  Conic  Sections,  of  which 
curves  they  had  discovered  all  the  principal  properties. 
But  it  is  curious  to  remark,  that  these  investigations, 
thus  pursued  at  first  as  mere  matters  of  curiosity  and 
intellectual  gratification,  were  destined,  two  thousand 
years  later,  to  play  a  very  important  part  in  establishing 
that  system  of  the  celestial  motions  which  succeeded  the 
Platonic  scheme  of  cycles  and  epicycles.  If  the  proper 
ties  of  the  conic  sections  had  not  been  demonstrated  by 
the  Greeks,  and  thus  rendered  familiar  to  the  mathe 
maticians  of  succeeding  ages,  Kepler  would  probably 


not  have  been  able  to  discover  those  laws  respecting  the 
orbits  and  motions  of  the  planets  which  were  the  occa 
sion  of  the  greatest  revolution  that  ever  happened  in 
the  history  of  science. 

4.  The  Arabians,    who,    as  I  have  elsewhere  said, 
added  little  of  their  own  to  the  stores  of  science  which 
they  received  from  the  Greeks,  did  however  make  some 
very  important  contributions  in  those  portions  of  pure 
mathematics  which  are  subservient  to  astronomy.     Their 
adoption  of  the  Indian  mode  of  computation  by  means 
of  the  Ten  Digits,  1,  2,  3,  4,  5,  G,  7,  8,  9,  0,  and  by  the 
method  of  Local  Values,  instead  of  the  cumbrous  sexa 
gesimal  arithmetic  of  the  Greeks,  was  an  improvement 
by  which  the  convenience  and  facility  of  numerical  cal 
culations  were  immeasurably  augmented.     The  Arabians 
also  rendered  several  of  the  processes  of  trigonometry 
much  more  commodious,  by  using  the  Sine  of  an  arc 
instead  of  the  Chord ;  an  improvement  which  Albateg- 
nius  appears  to  claim  for  himself'-';  and  by  employing 
also   the    Tangents  of  arcs,  or,   as  they  called  themf, 
upright  shadows. 

5.  The  constant  application  of  mathematical  know 
ledge  to  the  researches  of  Astronomy,  and  the  mutual 
influence  of  each  science  on  the  progress  of  the  other, 
has  been  still  more  conspicuous  in  modern  times.     New 
ton's  Method  of  Prime  and  Ultimate  Ratios,  which  we 
have  already  noticed  as  the  first  correct  exposition  of 
the  doctrine  of  a  Limit,  is  stated  in  a  series  of  Lemmas, 
or  preparatory  theorems,  prefixed  to  his  Treatise  on  the 
System  of  the    World.     Both  the   properties  of  curve 
lines  and  the   doctrines  concerning  force  and    motion, 
which  he  had  to  establish,   required   that  the  common 
mathematical   methods    should  be  methodized  and  ex 
tended.     If  Newton  had  not  been  a  most  expert  and  in- 

*  Delambre,  Ast,  M.  A.,  p.  12.  +  I  hid.,  p.  17. 


ventive  mathematician,  as  well  as  a  profound  and  philo 
sophical  thinker,  he  could  never  have  made  any  one  of 
those  vast  strides  in  discovery  of  which  the  rapid  succes 
sion  in  his  work  strikes  us  with  wonder*.  And  if  we 
see  that  the  great  task  begun  by  him,  goes  on  more 
slowly  in  the  hands  of  his  immediate  successors,  and 
lingers  a  little  before  its  full  completion,  we  perceive 
that  this  arises,  in  a  great  measure,  from  the  defect  of 
the  mathematical  methods  then  used.  Newton's  syn 
thetical  modes  of  investigation,  as  we  have  elsewhere 
observed,  were  an  instrument  f,  powerful  indeed  in  his 
mighty  hand,  but  too  ponderous  for  other  persons  to 
employ  with  effect.  The  countrymen  of  Newton  clung 
to  it  the  longest,  out  of  veneration  for  their  master ;  and 
English  cultivators  of  physical  astronomy  were,  on  that 
very  account,  left  behind  the  progress  of  mathematical 
science  in  France  and  Germany,  by  a  wide  interval, 
which  they  have  only  recently  recovered.  On  the  Conti 
nent,  the  advantages  offered  by  a  familiar  use  of  symbols, 
and  by  attention  to  their  symmetry  and  other  relations, 
were  accepted  without  reserve.  In  this  manner  the 
Differential  Calculus  of  Leibnitz,  which  was  in  its  origin 
and  signification  identical  with  the  Method  of  Fluxions 
of  Newton,  soon  surpassed  its  rival  in  the  extent  and 
generality  of  its  application  to  problems.  This  Calculus 
was  applied  to  the  science  of  mechanics,  to  which  it, 
along  with  the  symmetrical  use  of  co-ordinates,  gave  a 
new  form ;  for  it  was  soon  seen  that  the  most  difficult 
problems  might  in  general  be  reduced  to  finding  inte 
grals,  which  is  the  reciprocal  process  of  that  by  which 
differentials  are  found ;  so  that  all  difficulties  of  physical 
astronomy  were  reduced  to  difficulties  of  symbolical  cal 
culation,  these,  indeed,  being  often  sufficiently  stubborn. 
Clairaut,  Euler,  and  D'Alembert  employed  the  increased 
*  Hist.  Ind.  $c.,  R.  vn.  c.  ii.  t  //,.,  p.  175. 


resources  of  mathematical  science  upon   the  Theory  of 
the  Moon,  and  other  questions  relative  to  the  system  of 
the  world ;  and  thus  began  to  pursue  such  inquiries  in 
the  course  in  which  mathematicians  are  still  labouring 
up  to  the  present  day.     This  course  was  not  without  its 
checks  and  perplexities.     We  have  elsewhere  quoted* 
Clairaut's  expression  when  he  had  obtained  the   very 
complex  differential  equations  which  contain  the  solu 
tion  of  the  problem  of  the  moon's  motion :'  "  Now  inte 
grate  them  who  can !"    But  in  no  very  long  time  they 
were  integrated,  at  least  approximately ;  and  the  methods 
of  approximation   have  since   then   been  improved ;  so 
that  now,  with  a  due  expenditure  of  labour,  they  may  be 
carried  to  any  extent  which   is  thought  desirable.     If 
the  methods  of  astronomical  observation  should  here 
after  reach  a  higher  degree  of  exactness  than  they  now 
profess,  so  that  irregularities  in  the  motions  of  the  sun, 
moon,  and  planets,  shall  be  detected  which  at  present 
escape  us,  the  mathematical  part  of  the  theory  of  univer 
sal  gravitation  is  in  such  a  condition  that  it  can  soon  be 
brought  into  comparison  with  the  newly-observed  facts. 
Indeed  at  present  the  mathematical  theory  is  in  advance 
of  such  observations.     It  can  venture  to  suggest  what 
may  afterwards  be  detected,  as  well  as  to  explain  what 
has  already  been  observed.     This  has  happened  recently; 
for  Professor  Airy  has  calculated  the  law  and  amount 
of  an  inequality  depending  upon  the  mutual  attraction  of 
the  Earth  and  Venus ;  of  which  inequality  (so  small  is 
it,)  it  remains  to  be  determined  whether  its  effect  can  be 
traced  in  the  series  of  astronomical  observations. 

G.  As  the  influence  of  mathematics  upon  the  progress 
of  astronomy  is  thus  seen  in  the  cases  in  which  theory 
and  observation  confirm  each  other,  so  this  influence  ap 
pears  in  another  way,  in  the  very  few  cases  in  which  the 

*   Hist.  Ind.  Sc.,  R  vi.  c.  vi.  sect.  7- 


facts  have  not  been  fully  reduced  to  an  agreement  with 
theory.  The  most  conspicuous  case  of  this  kind  is  the 
state  of  our  knowledge  of  the  Tides.  This  is  a  portion 
of  astronomy :  for  the  Newtonian  theory  asserts  these 
curious  phenomena  to  be  the  result  of  the  attraction  of 
the  sun  and  moon.  Nor  can  there  be  any  doubt  that 
this  is  true,  as  a  general  statement ;  yet  the  subject  is 
up  to  the  present  time  a  blot  on  the  perfection  of  the 
theory  of  universal  gravitation ;  for  we  are  very  far  from 
being  able  in  this,  as  in  the  other  parts  of  astronomy,  to 
show  that  theory  will  exactly  account  for  the  time,  and 
magnitude,  and  all  other  circumstances  of  the  pheno 
menon  at  every  place  on  the  earth's  surface.  And  what 
is  the  portion  of  our  mathematics  which  is  connected 
with  this  solitary  signal  defect  in  astronomy  ?  It  is  the 
mathematics  of  the  Motion  of  Fluids ;  a  portion  in  which 
extremely  little  progress  has  been  made,  and  in  which  all 
the  more  general  problems  of  the  subject  have  hitherto 
remained  entirely  insoluble.  The  attempts  of  the  greatest 
mathematicians,  Newton,  Maclaurin,  Bernoulli,  Clairaut, 
Laplace,  to  master  such  questions,  all  involve  some  gra 
tuitous  assumption,  which  is  introduced  because  the 
problem  cannot  otherwise  be  mathematically  dealt  with : 
these  assumptions  confessedly  render  the  result  defective, 
and  how  defective,  it  is  hard  to  say.  And  it  was  pro 
bably  precisely  the  absence  of  a  theory  which  could  be 
reasonably  expected  to  agree  with  the  observations,  which 
made  Observations  of  this  very  curious  phenomenon,  the 
Tides,  to  be  so  much  neglected  as  till  very  recently  they 
were.  Of  late  years  such  observations  have  been  pur 
sued,  and  their  results  have  been  resolved  into  empirical 
laws,  so  that  the  rules  of  the  phenomena  have  been 
ascertained,  although  the  dependence  of  these  rules  upon 
the  lunar  and  solar  forces  has  not  been  shown.  Here 
then  we  have  a  portion  of  our  knowledge  relating  to 


facts  undoubtedly  dependent  upon  universal  gravitation, 
in  which  Observation  has  outstripped  Theory  in  her  pro 
gress,  and  is  compelled  to  wait  till  her  usual  companion 
overtakes  her.  This  is  a  position  of  which  Mathematical 
Theory  has  usually  been  very  impatient,  and  we  may 
expect  that  she  will  be  no  less  so  in  the  present  instance. 

7.  It  would  be  easy  to  show  from  the  history  of 
other  sciences,  for  example,  Mechanics  and  Optics,  how 
essential  the  cultivation  of  pure  mathematics  has  been  to 
their  progress.    The  parabola  was  already  familiar  among 
mathematicians  when  Galileo  discovered  that  it  was  the 
theoretical  path  of  a  Projectile ;  and  the  extension  and 
generalization  of  the  Laws  of  Motion  could  never  have 
been  effected,  unless  the  Differential  and  Integral  Cal 
culus  had  been  at  hand,  ready  to  trace  the  results  of  every 
hypothesis  which  could  be  made.    D'Alembert's  mode  of 
expressing  the  Third  Law  of  Motion  in  its  most  general 
form"''",  if  it  did  not  prove  the  law,  at  least  reduced  the 
application  of  it  to  analytical  processes  which  could  be 
performed  in  most  of  those  cases  in   which  they  were 
needed.     In  many  instances  the  demands  of  mechanical 
science  suggested  the  extension  of  the  methods  of  pure 
analysis.     The  problem  of  Vibrating  Strings  gave  rise  to 
the  Calculus  of  Partial  Differences,  which  was  still  fur 
ther  stimulated  by  its  application  to  the  motions  of  fluids 
and  other  mechanical  problems.     And  we  have  in  the 
writings  of  Lagrange  and  Laplace  other  instances  equally 
remarkable  of  new  analytical  methods,  to  \vhich  mecha 
nical  problems,  and  especially  cosmical  problems,  have 
given  occasion. 

8.  The  progress  of  Optics  as  a  science  has,  in  like 
manner,  been  throughout  dependent  upon  the  progress 
<>i'  pure  mathematics.     The  first  rise  of  geometry  was  fol- 

*  hid.  Sd.,  R  vi.  c.  vi.  sort.  7- 

VOL.  i.    w.  r.  M 

1()2  PHILOSOPHY    OI-:    Till-:    PUUK    SCIKNCKS. 

lowed  by  some  advances,  slight  ones  no  doubt,  in  the 
doctrine  of  Reflection  and  in  Perspective.  The  law  of 
Refraction  was  traced  to  its  consequences  by  means  of 
Trigonometry,  which  indeed  was  requisite  to  express  the 
law  in  a  simple  form.  The  steps  made  in  Optical  science 
by  Descartes,  Newton,  Euler,  and  Huyghens,  required 
the  geometrical  skill  which  those  philosophers  possessed. 
And  if  Young  and  Fresnel  had  not  been,  each  in  his 
peculiar  way,  persons  of  eminent  mathematical  endow 
ments,  they  would  not  have  been  able  to  bring  the 
Theory  of  Undulations  and  Interferences  into  a  condi 
tion  in  which  it  could  be  tested  by  experiments.  We 
may  see  how  unexpectedly  recondite  parts  of  pure  mathe 
matics  may  bear  upon  physical  science,  by  calling  to 
mind  a  circumstance  already  noticed  in  the  History  of 
Science* ; — that  Fresnel  obtained  one  of  the  most  curious 
confirmations  of  the  theory  (the  laws  of  Circular  Polar 
ization  by  reflection)  through  an  interpretation  of  an 
algebraical  expression,  which,  according  to  the  original 
conventional  meaning  of  the  symbols,  involved  an  im 
possible  quantity.  We  have  already  remarked,  that  in 
virtue  of  the  principle  of  the  generality  of  symbolical 
language,  such  an  interpretation  may  often  point  out 
some  real  and  important  analogy. 

9.  From  this  rapid  sketch  it  may  be  seen  how 
important  an  office  in  promoting  the  progress  of  the 
physical  sciences  belongs  to  mathematics.  Indeed  in 
the  progress  of  many  sciences,  every  step  has  been  so 
intimately  connected  with  some  advance  in  mathematics, 
that  we  can  hardly  be  surprized  if  some  persons  have 
considered  mathematical  reasoning  to  be  the  most  essen 
tial  part  of  such  sciences ;  and  have  overlooked  the  other 
elements  which  enter  into  their  formation.  How  erro- 
*  Hist.  Ind.  <Scz'.,  B.  ix.  c.  xiii.  sect.  2. 


neous  this  view  is  we  shall  best  see  by  turning  our 
attention  to  the  other  Ideas  besides  those  of  space,  num 
ber,  and  motion,  which  enter  into  some  of  the  most 
conspicuous  and  admired  portions  of  what  is  termed 
exact  science ;  and  by  showing  that  the  clear  and  distinct 
developement  of  such  Ideas  is  quite  as  necessary  to  the 
progress  of  exact  and  real  knowledge  as  an  acquaintance 
with  arithmetic  and  geometry. 

M  a 





IN  the  History  of  the  Sciences,  that  class  of  which  we 
here  speak  occupies  a  conspicuous  and  important  place ; 
coming  into  notice  immediately  after  those  parts  of  astro 
nomy  which  require  for  their  cultivation  merely  the 
ideas  of  space,  time,  motion,  and  number.  It  appears 
from  our  History,  that  certain  truths  concerning  the  equi 
librium  of  bodies  were  established  by  Archimedes  ; — that, 
after  a  long  interval  of  inactivity,  his  principles  were 
extended  and  pursued  further  in  modern  times : — and 
that  to  these  doctrines  concerning  equilibrium  and  the 
forces  which  produce  it,  (which  constitute  the  science 
Statics,}  were  added  many  other  doctrines  concerning 
the  motions  of  bodies,  considered  also  as  produced  by 
forces,  and  thus  the  science  of  Dynamics  was  produced. 
The  assemblage  of  these  sciences  composes  the  province 
of  Mechanics.  Moreover,  philosophers  have  laboured  to 
make  out  the  laws  of  the  equilibrium  of  fluid  as  well  as 
solid  bodies ;  and  hence  has  arisen  the  science  of  Hydro 
statics.  And  the  doctrines  of  Mechanics  have  been  found 
to  have  a  most  remarkable  bearing  upon  the  motions 
of  the  heavenly  bodies ;  with  reference  to  which,  indeed, 
they  were  at  first  principally  studied.  The  explanation 


of  those  cosmical  facts  by  means  of  mechanical  principles 
and  their  consequences,  forms  the  science  of  Physical 
Astronomy.  These  are  the  principal  examples  of  mecha 
nical  science ;  although  some  other  portions  of  Physics, 
as  Magnetism  and  Electrodynamics,  introduce  mecha 
nical  doctrines  very  largely  into  their  speculations. 

Now  in  all  these  sciences  we  have  to  consider  Forces. 
In  all  mechanical  reasonings  forces  enter,  either  as  pro 
ducing  motion,  or  as  prevented  from  doing  so  by  other 
forces.  Thus  force,  in  its  most  general  sense,  is  the  cause 
of  motion,  or  of  tendency  to  motion ;  and  in  order  to 
discover  the  principles  on  which  the  mechanical  sciences 
truly  rest,  we  must  examine  the  nature  and  origin  of 
our  knowledge  of  Causes. 

In  these  sciences,  however,  we  have  not  to  deal  with 
Cause  in  its  more  general  acceptation,  in  which  it  applies 
to  all  kinds  of  agency,  material  or  immaterial ; — to  the 
influence  of  thought  and  will,  as  well  as  of  bodily  pres 
sure  and  attractive  force.  Our  business  at  present  is 
only  with  such  causes  as  immediately  operate  upon 
matter.  We  shall  nevertheless,  in  the  first  place,  con 
sider  the  nature  of  Cause  in  its  most  general  form ;  and 
afterwards  narrow  our  speculations  so  as  to  direct  them 
specially  to  the  mechanical  sciences. 

OF    THE    IDEA    OF    CAUSE. 

1.  WE  see  in  the  world  around  us  a  constant  suc 
cession  of  causes  and  effects  connected  with  each  other. 
The  laws  of  this  connexion  we  learn  in  a  great  measure 
from  experience,  by  observation  of  the  occurrences  which 
present  themselves  to  our  notice,  succeeding  one  another. 


But  in  doing  this,  and  in  attending  to  this  succession  of 
appearances,  of  which  we  are  aware  by  means  of  our 
senses,  we  supply  from  our  own  minds  the  Idea  of  Cause. 
This  Idea,  as  we  have  already  shown  with  respect  to 
other  Ideas,  is  not  derived  from  experience,  but  has  its 
origin  in  the  mind  itself; — is  introduced  into  our  expe 
rience  by  the  active,  and  not  by  the  passive  part  of  our 

By  Cause  we  mean  some  quality,  power,  or  efficacy, 
by  which  a  state  of  things  produces  a  succeeding  state. 
Thus  the  motion  of  bodies  from  rest  is  produced  by  a 
cause  which  we  call  Force :  and  in  the  particular  case 
in  which  bodies  fall  to  the  earth,  this  force  is  termed 
Gravity.  In  these  cases,  the  Conceptions  of  Force  and 
Gravity  receive  their  meaning  from  the  Idea  of  Cause 
which  they  involve :  for  Force  is  conceived  as  the  Cause 
of  Motion.  That  this  Idea  of  Cause  is  not  derived  from 
experience,  we  prove  (as  in  former  cases)  by  this  con 
sideration  :  that  we  can  make  assertions,  involving  this 
idea,  which  are  rigorously  necessary  and  universal ; 
whereas  knowledge  derived  from  experience  can  only  be 
true  as  far  as  experience  goes,  and  can  never  contain  in 
itself  any  evidence  whatever  of  its  necessity.  We  assert 
that  "  Every  event  must  have  a  cause :"  and  this  proposi 
tion  we  know  to  be  true,  not  only  probably,  and  gene 
rally,  and  as  far  as  we  can  see :  but  we  cannot  suppose 
it  to  be  false  in  any  single  instance.  We  are  as  certain 
of  it  as  of  the  truths  of  arithmetic  or  geometry.  We 
cannot  doubt  that  it  must  apply  to  all  events  past  and 
future,  in  every  part  of  the  universe,  just  as  truly  as 
to  those  occurrences  which  we  have  ourselves  observed. 
What  causes  produce  what  effects; — what  is  the  cause 
of  any  particular  event ; — what  will  be  the  effect  of  any 
peculiar  process ; — these  are  points  on  which  experience 
may  enlighten  us.  Observation  and  experience  may  be 

OF   THE   IDEA    OF   CAUSE.  107 

requisite,  to  enable  us  to  judge  respecting  such  matters. 
But  that  every  event  has  some  cause,  Experience  cannot 
prove  any  more  than  she  can  disprove.  She  can  add 
nothing  to  the  evidence  of  the  truth,  however  often  she 
may  exemplify  it.  This  doctrine,  then,  cannot  have  been 
acquired  by  her  teaching ;  and  the  Idea  of  Cause,  which 
the  doctrine  involves,  and  on  which  it  depends,  cannot 
have  come  into  our  minds  from  the  region  of  observa 

2.  That  we  do,  in  fact,  apply  the  Idea  of  Cause  in  a 
more  extensive  manner  than  could  be  justified,  if  it  were 
derived  from  experience  only,  is  easily  shown.    For  from 
the  principle  that  everything  must  have  a  cause,  we  not 
only  reason  concerning  the  succession  of  the  events  which 
occur  in  the  progress  of  the  world,  and  which  form  the 
course  of  experience ;  but  we  infer  that  the  world  itself 
must  have  a  cause ; — that  the  chain  of  events  connected 
by  common   causation,  must  have  a  First  Cause  of  a 
nature  different  from  the  events  themselves.     This  we 
are  entitled  to  do,  if  our  Idea  of  Cause  be  independent  of, 
and  superior  to,  experience :  but  if  we  have  no  Idea  of 
Cause  except  such  as  we  gather  from  experience,  this 
reasoning  is  altogether  baseless  and  unmeaning. 

3.  Again  ;  by  the  use  of  our  powers  of  observation, 
we  are  aware  of  a  succession  of  appearances  and  events. 
But  none  of  our  senses  or  powers  of  external  observa 
tion  can  detect  in  these  appearances  the  power  or  quality 
which  we  call  Cause.     Cause  is  that  which  connects  one 
event  with  another ;  but  no  sense  or  perception  discloses 
to  us,  or  can  disclose,  any  connexion  among  the  events 
which  we  observe.    We  see  that  one  occurrence  follows 
another,  but  we  can  never  see  anything  which  shows  that 
one  occurrence  must  follow  another.     We  have  already 
noticed*,  that  this  truth  has  been  urged  by  nictaphv- 

Book  I.,  chap.  xiii. 


sicians  in  modern  times,  and  generally  assented  to  by 
those  who  examine  carefully  the  connexion  of  their  own 
thoughts.  The  arguments  are,  indeed,  obvious  enough. 
One  ball  strikes  another  and  causes  it  to  move  forwards. 
But  by  what  compulsion?  Where  is  the  necessity?  If 
the  mind  can  see  any  circumstance  in  this  case  which 
makes  the  result  inevitable,  let  this  circumstance  be 
pointed  out.  But,  in  fact,  there  is  no  such  discoverable 
necessity ;  for  we  can  conceive  this  event  not  to  take 
place  at  all.  The  struck  ball  may  stand  still,  for  aught 
we  can  see.  "  But  the  laws  of  motion  will  not  allow  it 
to  do  so."  Doubtless  they  will  not.  But  the  laws  of 
motion  are  learnt  from  experience,  and  therefore  can 
prove  no  necessity.  Why  should  not  the  laws  of  motion 
be  other  than  they  are?  Are  they  necessarily  true? 
That  they  are  necessarily  such  as  do  actually  regulate  the 
impact  of  bodies,  is  at  least  no  obvious  truth ;  and  there 
fore  this  necessity  cannot  be,  in  common  minds,  the 
ground  of  connecting  the  impact  of  one  ball  with  the 
motion  of  another.  And  assuredly,  if  this  fail,  no  other 
ground  of  such  necessary  connexion  can  be  shown.  In 
this  case,  then,  the  events  are  not  seen  to  be  necessarily 
connected.  But  if  this  case,  where  one  ball  moves  another 
by  impulse,  be  not  an  instance  of  events  exhibiting  a 
necessary  connexion,  we  shall  look  in  vain  for  any  ex 
ample  of  such  a  connexion.  There  is,  then,  no  case  in 
which  events  can  be  observed  to  be  necessarily  con 
nected  :  our  idea  of  causation,  which  implies  that  the 
event  is  necessarily  connected  with  the  cause,  cannot  be 
derived  from  observation. 

4.  But  it  may  be  said,  we  have  not  any  such  Idea  of 
Cause,  implying  necessary  connexion  with  effect,  and  a 
quality  by  which  this  connexion  is  produced.  We  see 
nothing  but  the  succession  of  events ;  and  by  muse  we 
mean  nothing  but  a  certain  succession  of  events; — name- 

OF    THE    IDEA    OF    CAUSE.  1  GO 

ly,  a  constant,  unvarying  succession.  Cause  and  effect 
are  only  two  events  of  which  the  second  invariably 
follows  the  first.  We  delude  ourselves  when  we  ima 
gine  that  our  idea  of  causation  involves  anything  more 
than  this. 

To  this  I  reply  by  asking,  what  then  is  the  meaning 
of  the  maxim  above  quoted,  and  allowed  by  all  to  be 
universally  and  necessarily  true,  that  every  event  must 
have  a  cause  ?  Let  us  put  this  maxim  into  the  language 
of  the  explanation  just  noticed ;  and  it  becomes  this  :— 
"Every  event  must  have  a  certain  other  event  invariably 
preceding  it."  But  why  must  it?  Where  is  the  neces 
sity  ?  Why  must  like  events  always  be  preceded  by  like, 
except  so  far  as  other  events  interfere  ?  That  there  is 
such  a  necessity,  no  one  can  doubt.  All  will  allow  that 
if  a  stone  ascend  because  it  is  thrown  upwards  in  one 
case,  a  stone  which  ascends  in  another  case  has  also 
been  thrown  upwards,  or  has  undergone  some  equi 
valent  operation.  All  will  allow  that  in  this  sense, 
every  kind  of  event  must  have  some  other  specific  kind 
of  event  preceding  it.  But  this  turn  of  men's  thoughts 
shows  that  they  see  in  events  a  connexion  which  is  not 
mere  succession.  They  see  in  cause  and  effect,  not 
merely  what  does,  often  or  always,  precede  and  follow, 
but  what  must  precede  and  follow.  The  events  are  not 
only  conjoined,  they  are  connected.  The  cause  is  more 
than  the  prelude,  the  effect  is  more  than  the  sequel,  of 
the  fact.  The  cause  is  conceived  not  as  a  mere  occa 
sion  ;  it  is  a  power,  an  efficacy,  which  has  a  real  ope 

5.  Thus  we  have  drawn  from  the  maxim,  that  Every 
Kffect  must  have  a  Cause,  arguments  to  show  that  we 
have  an  Idea  of  Cause  which  is  not  borrowed  from  expe 
rience,  and  which  involves  more  than  mere  succession. 
Similar  arguments  might  be  derived  from  any  other 


maxims  of  universal  and  necessary  validity,  which  we 
can  obtain  concerning  Cause :  as,  for  example,  the  max 
ims  that  Causes  are  measured  by  their  Effects,  and  that 
Reaction  is  equal  and  opposite  to  Action.  These  maxims 
we  shall  soon  have  to  examine  ;  but  we  may  observe  here, 
that  the  necessary  truth  which  belongs  to  them,  shows 
that  they,  and  the  Ideas  which  they  involve,  are  not  the 
mere  fruits  of  observation;  while  their  meaning,  including, 
as  it  does,  something  quite  different  from  the  mere  con 
ception  of  succession  of  events,  proves  that  such  a  con 
ception  is  far  from  containing  the  whole  import  and 
signification  of  our  Idea  of  Cause. 

The  progress  of  the  opinions  of  philosophers  on  the 
points  discussed  in  this  chapter,  has  been  one  of  the 
most  remarkable  parts  of  the  history  of  Metaphysics  in 
modern  times :  and  I  shall  therefore  briefly  notice  some 
of  its  features. 



1.  TOWARDS  the  end  of  the  seventeenth  century  there 
existed  in  the  minds  of  many  of  the  most  vigorous  and 
active  speculators  of  the  European  literary  world,  a  strong 
tendency  to  ascribe  the  whole  of  our  Knowledge  to  the 
teaching  of  Experience.  This  tendency,  with  its  conse 
quences,  including  among  them  the  reaction  which  was 
produced  when  the  tenet  had  been  pushed  to  a  length 
manifestly  absurd,  has  exercised  a  very  powerful  in 
fluence  upon  the  progress  of  metaphysical  doctrines  up 
to  the  present  time.  I  proceed  to  notice  some  of  the 
most  prominent  of  the  opinions  which  have  thus  ob- 

OPINIONS    RESPECTING    THE    IDEA    OF    CAUSE.       171 

tained  prevalence  among  philosophers,  so  far  as  the  Idea 
of  Cause  is  concerned. 

Locke  was  one  of  the  metaphysicians  who  produced 
the  greatest  effect  in  diffusing  this  opinion,  of  the  exclu 
sive  dependence  of  our  knowledge  upon  experience. 
Agreeably  to  this  general  system,  he  taught  *  that  our 
ideas  of  Cause  and  Effect  are  got  from  observation  of 
the  things  about  us.  Yet  notwithstanding  this  tenet  of 
his,  he  endeavoured  still  to  employ  these  ideas  in  rea 
soning  on  subjects  which  are  far  beyond  all  limits  of 
experience :  for  he  professed  to  prove,  from  our  idea  of 
Causation,  the  existence  of  the  Deity  f. 

Hume  noticed  this  obvious  inconsistency;  but  declared 
himself  unable  to  discover  any  remedy  for  a  defect  so 
fatal  to  the  most  important  parts  of  our  knowledge.     He 
could  see,  in  our  belief  of  the  succession  of  cause  and 
effect,  nothing  but  the  habit  of  associating  in  our  minds 
what  had  often  been  associated  in  our  experience.     He 
therefore  maintained   that  we  could  not,  with    logical 
propriety,  extend  our  belief  of  such  a  succession  to  cases 
entirely  distinct  from  all  those  of  which  our  experience 
consisted.    We  see,  he  said,  an  actual  conjunction  of  two 
events ;  but  we  can  in  no  way  detect  a  necessary  con 
nexion  ;  and  therefore  we  have  no  means  of  inferring 
cause  from  effect,  or  effect  from  causej.     The  only  way 
in  which  we  recognize  Cause  and  Effect  in  the  field  of 
our  experience,  is  as  an  unfailing  Sequence :  we  look  in 
vain  for  anything  which  can  assure  us  of  an  infallible 
Consequence.     And  since  experience  is  the  only  source 
of  our  knowledge,  we   cannot   with  any  justice  assert 
that  the  world  in  which  we  live  must  necessarily  have 
had  a  cause. 

2.  This  doctrine,  taken  in  conjunction  with  the  known 

*   Essay  on  the  Human  Understanding,  B.  it.  c  xxvi.      t  B.  iv.  c.  x. 
+  Hume's  Phil,  of  the  Human  Mind,  Vol.  I.  p.  94. 

172        PHILOSOPHY    OF    Til  1C    MECHANICAL    SCIENCES. 

skepticism  of  its  author  on  religious  points,  produced  a 
considerable  fermentation  in  the  speculative  world.  The 
solution  of  the  difficulty  thus  thrown  before  philosophers, 
was  by  no  means  obvious.  It  was  vain  to  endeavour  to 
find  in  experience  any  other  property  of  a  Cause,  than  a 
constant  sequence  of  the  effect.  Yet  it  was  equally  vain 
to  try  to  persuade  men  that  they  had  no  idea  of  Cause ; 
or  even  to  shake  their  belief  in  the  cogency  of  the  fami 
liar  arguments  concerning  the  necessity  of  an  original 
cause  of  all  that  is  and  happens.  Accordingly  these 
hostile  and  apparently  irreconcilable  doctrines, — the  in 
dispensable  necessity  of  a  cause  of  every  event,  and  the 
impossibility  of  our  knowing  such  a  necessity, — were  at 
last  allowed  to  encamp  side  by  side.  Reid,  Beattie,  and 
others,  formed  one  party,  who  showed  how  widely  and 
constantly  the  idea  of  a  cause  pervades  all  the  processes 
of  the  human  mind  :  while  another  sect,  including  Brown, 
and  apparently  Stewart,  maintained  that  this  idea  is 
always  capable  of  being  resolved  into  a  constant  se 
quence  ;  and  these  latter  reasoners  tried  to  obviate  the 
dangerous  and  shocking  inferences  which  some  persons 
might  try  to  draw  from  their  opinion,  by  declaring  the 
maxim  that  "Every  event  must  have  a  cause,"  to  be  an 
instinctive  law  of  belief,  or  a  fundamental  principle  of 
the  human  mind*. 

3.  While  this  series  of  discussions  was  going  on  in 
Britain,  a  great  metaphysical  genius  in  Germany  was 
unravelling  the  perplexity  in  another  way.  Kant's  spe 
culations  originated,  as  he  informs  us,  in  the  trains  of 
thought  to  which  Hume's  writings  gave  rise ;  and  the 
Kritik  der  Reinen  Vernunft,  or  Examination  of  the 
Pure  Reason,  was  published  in  1787,  with  the  view  of 
showing  the  true  nature  of  our  knowledge. 

*  Stewart's  Active   Powers,   Vol.  i.   p.  347-      Brown's   Lectures, 
Vol.  i.  p.  115. 

OPINIONS    RESPECTING    THE    IDEA    OF    CAUSE.          1  7\] 

Kant's  solution  of  the  difficulties  just  mentioned 
differs  materially  from  that  above  stated.  According  to 
Brown"-,  succession  observed  and  cause  inferred, — the 
memory  of  past  conjunctions  of  events  and  the  belief  of 
similar  future  conjunctions, — are  facts,  independent,  so 
far  as  we  can  discover,  but  inseparably  combined  by  a 
law  of  our  mental  nature.  According  to  Kant,  causality 
is  an  inseparable  condition  of  our  experience :  a  con 
nexion  in  events  is  requisite  to  our  apprehending  them  as 
events.  Future  occurrences  must  be  connected  by  causa 
tion  as  the  past  have  been,  because  we  cannot  think  of 
past,  present,  and  future,  without  such  connexion.  We 
cannot  fix  the  mind  upon  occurrences,  without  including 
these  occurrences  in  a  series  of  causes  and  effects.  The 
relation  of  Causation  is  a  condition  under  which  we 
think  of  events,  as  the  relations  of  space  are  a  condition 
under  which  we  see  objects. 

4.  On  a  subject  so  abstruse,  it  is  not  easy  to  make 
our  distinctions  very  clear.  Some  of  Brown's  illustrations 
appear  to  approach  very  near  to  the  doctrine  of  Kant. 
Thus  he  saysf,  "The  form  of  bodies  is  the  relation  of 
their  elements  to  each  other  in  space, — the  power  of 
bodies  is  their  relation  to  each  other  in  time."  Yet  not 
withstanding  such  approximations  in  expression,  the 
Kantian  doctrine  appears  to  be  different  from  the  views 
of  Stewart  and  Brown,  as  commonly  understood.  Ac 
cording  to  the  Scotch  philosophers,  the  cause  and  the 
effect  are  two  things,  connected  in  our  minds  by  a  law 
of  our  nature.  But  this  view  requires  us  to  suppose  that 
we  can  conceive  the  law  to  be  absent,  and  the  course  of 
events  to  be  unconnected.  If  we  can  understand  what  is 
the  special  force  of  this  law,  we  must  be  able  to  imagine 
what  the  case  would  be  if  the  law  were  non-existing.  We 
must  be  able  to  conceive  a  mind  which  does  not  connect 
*  Led..  Vol.  i.  ]>.  114.  t  Led.,  i.  ]>.  127. 


effects  with  causes.  The  Kantian  doctrine,  on  the  other 
hand,  teaches  that  we  cannot  imagine  events  liberated 
from  the  connexion  of  cause  and  effect :  this  connexion  is 
a  condition  of  our  conceiving  any  real  occurrences :  we 
cannot  think  of  a  real  sequence  of  things,  except  as  in 
volving  the  operation  of  causes.  In  the  Scotch  system, 
the  past  and  the  future  are  in  their  nature  independent, 
but  bound  together  by  a  rule ;  in  the  German  system, 
they  share  in  a  common  nature  and  mutual  relation,  by 
the  act  of  thought  which  makes  them  past  and  future. 
In  the  former  doctrine  cause  is  a  tie  which  binds ;  in  the 
latter  it  is  a  character  which  pervades  and  shapes  events. 
The  Scotch  metaphysicians  only  assert  the  universality 
of  the  relation ;  the  German  attempts  further  to  explain 
its  necessity. 

This  being  the  state  of  the  case,  such  illustrations  as 
that  of  Dr.  Brown  quoted  above,  in  which  he  represents 
cause  as  a  relation  of  the  same  kind  with  form,  do  not 
appear  exactly  to  fit  his  opinions.  Can  the  relations  of 
figure  be  properly  said  to  be  connected  with  each  other 
by  a  law  of  our  nature,  or  a  tendency  of  our  mental  con 
stitution?  Can  we  ascribe  it  to  a  law  of  our  thoughts, 
that  we  believe  the  three  angles  of  a  triangle  to  be  equal 
to  two  right  angles?  If  so,  we  must  give  the  same 
reason  for  our  belief  that  two  straight  lines  cannot 
inclose  a  space ;  or  that  three  and  two  are  five.  But 
will  any  one  refer  us  to  an  ultimate  law  of  our  consti 
tution  for  the  belief  that  three  and  two  are  five  ?  Do 
we  not  see  that  they  are  so,  as  plainly  as  we  see  that 
they  are  three  and  two  ?  Can  we  imagine  laws  of  our 
constitution  abolished,  so  that  three  and  two  shall  make 
something  different  from  five ; — so  that  an  inclosed  space 
shall  lie  between  two  straight  lines ; — so  that  the  three 
angles  of  a  plane  triangle  shall  be  greater  than  two 
right  angles?  We  cannot  conceive  this.  If  the  mini- 

OPINIONS    RESPECTING    THE    IDEA    OF    CAUSE.       1  7 "> 

bers  are  three  and  two ;  if  the  lines  are  straight ;  if  the 
triangle  is  a  rectilinear  triangle,  the  consequences  are 
inevitable.  We  cannot  even  imagine  the  contrary.  We 
do  not  want  a  law  to  direct  that  things  should  be  what 
they  are.  The  relation,  then,  of  cause  and  effect,  being 
of  the  same  kind  as  the  necessary  relations  of  figure  and 
number,  is  not  properly  spoken  of  as  established  in  our 
minds  by  a  special  law  of  our  constitution :  for  we  reject 
that  loose  and  inappropriate  phraseology  which  speaks 
of  the  relations  of  figure  and  number  as  "  determined  by 
laws  of  belief." 

5.  In  the  present  work,  we  accept  and  adopt,  as  the 
basis  of  our  inquiry  concerning  our  knowledge,  the  exist 
ence  of  necessary  truths  concerning  causes,  as  there  exist 
necessary  truths  concerning   figure  and   number.     We 
find  such  truths  universally  established  and  assented  to 
among  the  cultivators  of  science,  and  among  speculative 
men  in  general.     All  mechanicians  agree  that  reaction 
is  equal  and  opposite  to  action,  both  when  one    body 
presses  another,  and  when  one  body  communicates  mo 
tion  to  another.     All  reasoners  join  in  the  assertion,  not 
only  that  every  observed  change  of  motion  has  had  a 
cause,  but  that  every  change  of  motion   must   have  a 
cause.     Here  we  have  certain    portions  of  substantial 
and  undoubted  knowledge.     Now  the  essential  point  in 
the  view  which  we  must  take  of  the  idea  of  cause  is 
this, — that  our  view  must  be  such  as  to  form  a  solid 
basis  for  our  knowledge.     We  have,  in  the*  Mechanical 
Sciences,  certain  universal  and  necessary  truths  on  the 
subject  of  causes.     Now  any  view  which  refers  our  be 
lief  in  causation  to  mere  experience  or  habit,  cannot 
explain  the   possibility  of  such  necessary  truths,   since 
experience  and  habit  can  never  lead  to  a  perception  of 
necessary  connexion.     But  a  view  which  teaches  us  to 
acknowledge  axioms  concerning  cause,  as  we  acknow- 


ledge  axioms  concerning  space,  will  lead  us  to  look  upon 
the  science  of  mechanics  as  equally  certain  and  univer 
sal  with  the  science  of  geometry ;  and  will  thus  mate 
rially  affect  our  judgment  concerning  the  nature  and 
claims  of  our  scientific  knowledge. 

Axioms  concerning  Cause,  or  concerning  Force, 
which  as  we  shall  see,  is  a  modification  of  Cause,  will 
flow  from  an  Idea  of  Cause,  just  as  axioms  concerning 
space  and  number  flow  from  the  ideas  of  space  and  num 
ber  or  time.  And  thus  the  propositions  which  con 
stitute  the  science  of  Mechanics  prove  that  we  possess 
an  idea  of  cause,  in  the  same  sense  in  which  the  propo 
sitions  of  geometry  and  arithmetic  prove  our  possession 
of  the  ideas  of  space  and  of  time  or  number. 

6.  The  idea  of  cause,  like  the  ideas  of  space  and 
time,  is  a  part  of  the  active  powers  of  the  mind.  The 
relation  of  cause  and  effect  is  a  relation  or  condition 
under  which  events  are  apprehended,  which  relation  is 
not  given  by  observation,  but  supplied  by  the  mind  itself. 
According  to  the  views  which  explain  our  apprehension 
of  cause  by  reference  to  habit,  or  to  a  supposed  law  of 
our  mental  nature,  causal  connexion  is  a  consequence  of 
agencies  which  the  mind  passively  obeys ;  but  according 
to  the  view  to  which  we  are  led,  this  connexion  is  a 
result  of  faculties  which  the  mind  actively  exercises. 
And  thus  the  relation  of  cause  and  effect  is  a  condition 
of  our  apprehending  successive  events,  a  part  of  the 
mind's  constant  and  universal  activity,  a  source  of  neces 
sary  truths ;  or,  to  sum  all  this  in  one  phrase,  a  Funda 
mental  Idea. 





1 .  Causes  are  abstract  Conceptions. — WE  have  now 
to  express,  as  well  as  we  can,  the  fundamental  character 
of  that  Idea  of  Cause,  of  which  we  have  just  proved  the 
existence.  This  may  be  done,  at  least  for  purposes  of 
reasoning,  in  this  as  in  former  instances,  by  means  of 
axioms.  I  shall  state  the  principal  axioms  which  belong 
to  this  subject,  referring  the  reader  to  his  own  thoughts 
for  the  axiomatic  evidence  which  belongs  to  them. 

But  I  must  first  observe,  that  in  order  to  express 
general  and  abstract  truths  concerning  cause  and  effect, 
these  terms,  cause  and  effect,  must  be  understood  in  a 
general  and  abstract  manner.  When  one  event  gives  rise 
to  another,  the  first  event  is,  in  common  language,  often 
called  the  cause,  and  the  second  the  effect.  Thus  the 
meeting;  of  two  billiard  balls  may  be  said  to  be  the 


cause  of  one  of  them  turning  aside  out  of  the  path  in 
which  it  was  moving.  For  our  present  purposes,  how 
ever,  we  must  not  apply  the  term  cause  to  such  occur 
rences  as  this  meeting  and  turning,  but  to  a  certain 
conception,  force,  abstracted  from  all  such  special  events, 
and  considered  as  a  quality  or  property  by  which  one 
body  affects  the  motion  of  the  other.  And  in  like  man 
ner  in  other  cases,  cause  is_to  be  conceived  as  some 
abstract  quality,  power,  or  efficacy,  by  which  change  is 
produced ;  a  quality  not  identical  \\itli  the  rvniK  but 
disclosed  by  means  of  them.  Not  only  is  this  abstract 
mode  of  conceiving  force  and  cause  useful  in  expressing 
the  fundamental  principles  of  science ;  but  it  supplies  us 
with  the  only  mode  by  which  such  principles  can  be 
•  VOL.  i.  w.  p.  N 


stated  in  a  general  manner,  and  made  to  lead  to  sub 
stantial  truth  and  real  knowledge. 

Understanding  cause,  therefore,  in  this  sense,  we 
proceed  to  our  Axioms. 

2.  First  Axiom.     Nothing  can  take  place  without  a 

Every  event,  of  whatever  kind,  must  have  a  Cause  in 
the  sense  of  the  term  which  we  have  just  indicated;  and 
that  it  must,  is  a  universal  and  necessary  proposition  to 
which  we  irresistibly  assent  as  soon  as  it  is  understood. 
We  believe  each  appearance  to  come  into  existence, — 
we  conceive  every  change  to  take  place, — not  only  with 
something  preceding  it,  but  something  by  which  it  is  made 
to  be  what  it  is.  An  effect  without  a  cause ; — an  event 
without  a  preceding  condition  involving  the  efficacy  by 
which  the  event  is  produced; — are  suppositions  which  we 
cannot  for  a  moment  admit.  That  the  connexion  of  effect 
with  cause  is  universal  and  necessary,  is  a  universal  and 
constant  conviction  of  mankind.  It  persists  in  the  minds 
of  all  men,  undisturbed  by  all  the  assaults  of  sophistry 
and  skepticism;  and,  as  we  have  seen  in  the  last  chapter, 
remains  unshaken,  even  when  its  foundations  seem  to  be 
ruined.  This  axiom  expresses,  to  a  certain  extent,  our 
Idea  of  Cause ;  and  when  that  idea  is  clearly  appre 
hended,  the  axiom  requires  no  proof,  and  indeed  admits 
of  none  which  makes  it  more  evident.  That  notwith 
standing  its  simplicity,  it  is  of  use  in  our  speculations,  we 
shall  hereafter  see ;  but  in  the  first  place,  we  must  con 
sider  the  other  axioms  belonging  to  this  subject. 

3.  Second  Axiom.    Effects  are  proportional  to  their 
Causes.,  and  Causes  are  measured  ~by  their  Effects. 

We  have  already  said  that  cause  is  that  quality  or 
power,  in  the  circumstances  of  each  case,  by  which  the 
effect  is  produced ;  and  this  power,  an  abstract  property 
of  the  condition  of  things  to  which  it  belongs,  can  in 

AXIOMS  wincii  RELATE  TO  THE  IDEA  OF  CAUSE.    1 79 

no  way  fall  directly  under  the  cognizance  of  the  senses. 
Cause,  of  whatever  kind,  is  not  apprehended  as  including 
objects  and  events  which  share  its  nature  by  being  co-ex 
tensive  with  certain  portions  of  it,  as  space  and  time  are. 
It  cannot  therefore,  like  them,  bo  measured  by  repeti 
tion  of  its  own  parts,  as  space  is  measured  by  repetition 
of  inches,  and  time  by  repetition  of  minutes.  Causes  may 
be  greater  or  less ;  as,  for  instance,  the  force  of  a  man  is 
greater  than  the  force  of  a  child.  But  how  much  is  the 
one  greater  than  the  other  ?  How  are  we  to  compare 
the  abstract  conception,  force,  in  such  cases  as  these  ? 

To  this,  the  obvious  and  only  answer  is,  that  we  must 
compare  causes  by  means  of  their  effects ; — that  we  must 
compare  force  by  something  which  force  can  do.  The 
child  can  lift  one  fagot;  the  man  can  lift  ten  such  fagots: 
we  have  here  a  means  of  comparison.  And  whether  or 
not  the  rule  is  to  be  applied  in  this  manner,  that  is,  by 
the  number  of  the  things  operated  on,  (a  question  which 
we  shall  have  to  consider  hereafter,)  it  is  clear  that  this 
form  of  rule,  namely,  a  reference  to  some  effect  or  other 
as  our  measure,  is  the  right,  because  the  only  possible 
form.  The  cause  determines  the  effect.  The  cause  being 
the  same,  the  effect  must  be  the  same.  The  connexion 
of  the  two  is  governed  by  a  fixed  and  inviolable  rule. 
It  admits  of  no  ambiguity.  Every  degree  of  intensity 
in  the  cause  has  some  peculiar  modification  of  the  effect 
corresponding  to  it.  Hence  the  effect  is  an  unfailing 
index  of  the  amount  of  the  cause ;  and  if  it  be  a  mea 
surable  effect,  gives  a  measure  of  the  cause.  We  can 
have  no  other  measure ;  but  we  need  no  other,  for  this 
is  exact,  sufficient,  and  complete. 

It  may  be  said,  that  various  effects  are  produced  by 
the  same  cause.  The  sun's  heat  melts  wax  and  expands 
quicksilver.  The  force  of  gravity  causes  bodies  to  move 
downwards  if  they  are  free,  and  to  press  down  upon  their 



supports  if  they  are  supported.  Which  of  the  effects  is  to 
be  taken  as  the  measure  of  heat,  or  of  gravity,  in  these 
cases  ?  To  this  we  reply,  that  if  we  had  merely  different 
states  of  the  same  cause  to  compare,  any  of  the  effects 
might  be  taken.  The  sun's  heat  on  different  days  might 
be  measured  by  the  expansion  of  quicksilver,  or  by  the 
quantity  of  wax  melted.  The  force  of  gravity,  if  it  were 
different  at  different  places,  might  be  measured  by  the 
spaces  through  which  a  given  weight  would  bend  an 
elastic  support,  or  by  the  spaces  through  which  a  body 
would  fall  in  a  given  time.  All  these  measures  are  con 
sistent  with  the  general  character  of  our  idea  of  cause. 

4.  Limitation  of  the  Second  Axiom. — But  there  may 
be  circumstances  in  the  nature  of  the  case  which  may 
further  determine  the  kind  of  effect  which  we  must  take 
for  the  measure  of  the  cause.  For  example,  if  causes 
are  conceived  to  be  of  such  a  nature  as  to  be  capable  of 
addition,  the  effects  taken  as  their  measure  must  conform 
to  this  condition.  This  is  the  case  with  mechanical 
causes.  The  weights  of  two  bodies  are  the  causes  of  the 
pressure  which  they  exert  downwards  ;  and  these  weights 
are  capable  of  addition.  The  weight  of  the  two  is  the 
sum  of  the  weight  of  each.  We  are  therefore  not  at 
liberty  to  say  that  weights  shall  be  measured  by  the 
spaces  through  which  they  bend  a  certain  elastic  support, 
except  we  have  first  ascertained  that  the  whole  weight 
bends  it  through  a  space  equal  to  the  sum  of  the  inflec 
tions  produced  by  the  separate  weights.  Without  this 
precaution,  we  might  obtain  inconsistent  results.  Two 
weights,  each  of  the  magnitude  0  as  measured  by  their 
effects,  might,  if  we  took  the  inflections  of  a  spring  for 
the  effects,  be  together  equal  to  5  or  to  7  by  the  same 
kind  of  measurement.  For  the  inflection  produced  by 
two  weights  of  3  might,  for  aught  we  can  see  before 
hand,  be  more  or  less  than  twice  as  great  as  the  inflection 

AXIOMS  WHICH    RELATE    TO    THE    IDEA    OK    CAUSE.       181 

produced  by  one  weight  of  3.  That  forces  are  capable  of 
addition,  is  a  condition  which  limits,  and,  as  we  shall  see, 
in  some  cases  rigorously  fixes,  the  kind  of  effects  which 
are  to  be  taken  as  their  measures. 

Causes  which  are  thus  capable  of  addition  are  to  be 
measured  by  the  repeated  addition  of  equal  quantities. 
Two  such  causes  are  equal  to  each  other  when  they  pro 
duce  exactly  the  same  effect.  So  far  our  axiom  is  applied 
directly.  But  these  two  causes  can  be  added  together ; 
and  being  thus  added,  they  are  double  of  one  of  them ; 
a*nd  the  cause  composed  by  addition  of  three  such,  is 
three  times  as  great  as  the  first ;  and  so  on  for  any  mea 
sure  whatever.  By  this  means,  and  by  this  means  only, 
we  have  a  complete  and  consistent  measure  of  those 
causes  which  are  so  conceived  as  to  be  subject  to  this 
condition  of  being  added  and  multiplied. 

Causes  are,  in  the  present  chapter,  to  be  understood 
in  the  widest  sense  of  the  term ;  and  the  axiom  now 
under  our  consideration  applies  to  them,  whenever  they 
are  of  such  a  nature  as  to  admit  of  any  measure  at  all. 
But  the  cases  which  we  have  more  particularly  in  view 
are  mechanical  causes,  the  causes  of  the  motion  and  of 
the  equilibrium  of  bodies.  In  these  cases,  forces  are  con 
ceived  as  capable  of  addition ;  and  what  has  been  said  of 
the  measure  of  causes  in  such  cases,  applies  peculiarly  to 
mechanical  forces.  Two  weights,  placed  together,  may 
be  considered  as  a  single  weight,  equal  to  the  sum  of  the 
two.  Two  pressures,  pushing  a  body  in  the  same  direc 
tion  at  the  same  point,  are  identical  in  all  respects  with 
some  single  pressure,  their  SUM,  pushing  in  like  manner; 
and  this  is  true  whether  or  not  they  put  the  body  in 
motion.  In  the  cases  of  mechanical  forces,  therefore,  we 
take  some  certain  effect,  velocity  generated  or  weight 
supported,  which  may  fix  the  unit  offeree:  and  we  then 
measure  all  other  forces  by  the  successive  repetition  of 


this  unit,  as  we  measure  all  spaces  by  the  successive 
repetition  of  our  unit  of  lineal  measure. 

But  these  steps  in  the  formation  of  the  science  of 
Mechanics  will  be  further  explained,  when  we  come  to 
follow  our  axioms  concerning  cause  into  their  application 
in  that  science.  At  present  we  have,  perhaps,  suffi 
ciently  explained  the  axiom  that  causes  are  measured 
by  their  effects,  and  we  now  proceed  to  a  third  axiom, 
also  of  great  importance. 

5.  Third  Axiom.  Reaction  is  equal  and  opposite  to 

In  the  case  of  mechanical  forces,  the  action  of  a 
cause  often  takes  place  by  an  operation  of  one  body 
upon  another ;  and  in  this  case,  the  action  is  always  and 
inevitably  accompanied  by  an  opposite  action.  If  I  press 
a  stone  with  my  hand,  the  stone  presses  my  hand  in 
return.  If  one  ball  strike  another  and  put  it  in  motion, 
the  second  ball  diminishes  the  motion  of  the  first.  In 
these  cases  the  operation  is  mutual ;  the  Action  is  ac 
companied  by  a  Reaction.  And  in  all  such  cases  the 
Reaction  is  a  force  of  exactly  the  same  nature  as  the 
Action,  exerted  in  an  opposite  direction.  A  pressure 
exerted  upon  a  body  at  rest  is  resisted  and  balanced  by 
another  pressure ;  when  the  pressure  of  one  body  puts 
another  in  motion,  the  body,  though  it  yields  to  the  force, 
nevertheless  exerts  upon  the  pressing  body  a  force  like 
that  which  it  suffers. 

Now  the  axiom  asserts  further,  that  this  Reaction 
is  equal,  as  well  as  opposite,  to  the  Action.  For  the 
Reaction  is  an  effect  of  the  Action,  and  is  determined  by 
it.  And  since  the  two,  Action  and  Reaction,  are  forces 
of  the  same  nature,  each  may  be  considered  as  cause 
and  as  effect ;  and  they  must,  therefore,  determine  each 
other  by  a  common  rule.  But  this  consideration  leads 
necessarily  to  their  equality :  for  since  the  rule  is  mutual, 


if  we  could  for  an  instant  suppose  the  Reaction  to  be 
less  than  the  Action,  we  must,  by  the  same  rule,  sup 
pose  the  Action  to  be  less  than  the  Reaction.  And  thus 
Action  and  Reaction,  in  every  such  case,  are  rigorously 
equal  to  each  other. 

It  is  easily  seen  that  this  axiom  is  not  a  proposition 
which  is,  or  can  be,  proved  by  experience ;  but  that  its 
truth  is  anterior  to  special  observation,  and  depends  on 
our  conception  of  Action  and  Reaction.  Like  our  other 
axioms,  this  has  its  source  in  an  Idea ;  namely,  the  Idea 
of  Cause,  under  that  particular  condition  in  which  cause 
and  effect  are  mutual.  The  necessary  and  universal 
truth  which  we  cannot  help  ascribing  to  the  axiom,  shows 
that  it  is  not  derived  from  the  stores  of  experience, 
which  can  never  contain  truths  of  this  character.  Ac 
cordingly,  it  was  asserted  with  equal  confidence  and 
generality  by  those  who  did  not  refer  to  experience  for 
their  principles,  and  by  those  who  did.  Leonicus  Tomreus, 
a  commentator  of  Aristotle,  whose  work  was  published 
in  1552,  and  therefore  at  a  period  when  no  right  opinions 
concerning  mechanical  reaction  were  current,  at  least 
in  his  school,  says,  in  his  remarks  on  the  Author's  Ques 
tions  concerning  the  communication  of  motion,  that 
"Reaction  is  equal  and  contrary  to  Action."  The  same 
principle  was  taken  for  granted  by  all  parties,  in  all  the 
controversies  concerning  the  proper  measure  of  force,  of 
which  we  shall  have  to  speak :  and  would  be  rigorously 
true,  as  a  law  of  motion,  whichever  of  the  rival  inter 
pretations  of  the  measure  of  the  term  ''Action"  we  were 
to  take. 

6.  Extent  of  the  Third  Axiom. — It  may  naturally  be 
asked  whether  this  third  Axiom  respecting  causation 
extends  to  any  other  cases  than  those  of  mechanical 
action,  since  the  notion  of  Cause  in  general  has  certainly 
a  much  wider  extent.  For  instance,  when  a  hot  bodv 


heats  a  cold  one,  is  there  necessarily  an  equal  reaction 
of  the  second  body  upon  the  first?  Does  the  snowball 
cool  the  boy's  hand  exactly  as  much  as  the  hand  heats 
the  snow  ?  To  this  we  reply,  that,  in  every  case  in  which 
one  body  acts  upon  another  by  its  physical  qualities,  there 
must  be  some  reaction.  No  body  can  affect  another 
without  being  itself  also  affected.  But  in  any  physical 
change  the  action  exerted  is  an  abstract  term  which  may 
be  variously  understood.  The  hot  hand  may  melt  a 
cold  body,  or  may  warm  it :  which  kind  of  effect  is  to 
be  taken  as  action?  This  remains  to  be  determined  by 
other  considerations. 

In  all  cases  of  physical  change  produced  by  one  body 
in  another,  it  is  generally  possible  to  assume  such  a 
meaning  of  action,  that  the  reaction  shall  be  of  the  same 
nature  as  the  action ;  and  when  this  is  done,  the  third 
axiom  of  causation,  that  reaction  is  equal  to  action,  is 
universally  true.  Thus  if  a  hot  body  heat  a  cold  one, 
the  change  may  be  conceived  as  the  transfer  of  a  certain 
substance,  heat  or  caloric,  from  the  first  body  to  the 
second.  On  this  supposition,  the  first  body  loses  just  as 
much  heat  as  the  other  gains ;  action  and  reaction  are 
equal.  But  if  the  reaction  be  of  a  different  kind  to  the 
action  we  can  no  longer  apply  the  axiom.  If  a  hot  body 
melt  a  cold  one,  the  latter  cools  the  former :  here,  then,  is 
reaction ;  but  so  long  as  the  action  and  reaction  are  stated 
in  this  form,  we  cannot  assert  any  equality  between  them. 

In  treating  of  the  secondary  mechanical  sciences,  we 
shall  see  further  in  what  way  we  may  conceive  the 
physical  action  of  one  body  upon  another,  so  that  the 
same  axioms  which  are  the  basis  of  the  science  of 
Mechanics  shall  apply  to  changes  not  at  first  sight  mani 
festly  mechanical. 

The  three  axioms  of  causation  which  we  have  now 
stated  are  the  fundamental  maxims  of  all  reasoning  con- 

AXIOMS   WHICH    RELATE  TO   THE    IDEA   OF   CAUSE.       185 

cerning  causes  as  to  their  quantities;  and  it  will  be 
shown  in  the  sequel  that  these  axioms  form  the  basis  of 
the  science  of  Mechanics,  determining  its  form,  extent, 
and  certainty.  We  must,  however,  in  the  first  place, 
consider  how  we  acquire  those  conceptions  upon  which 
the  axioms  now  established  are  to  be  employed. 



1.  Force.  —  WHEN  the  faculties  of  observation  and 
thought  are  developed  in  man,  the  idea  of  causation  is 
applied  to  those  changes  which  we  see  and  feel  in  the 
state  of  rest  and  motion  of  bodies  around  us.  And 
when  our  abstract  conceptions  are  thus  formed  and 
named,  we  adopt  the  term  Force,  and  use  it  to 
denote  that  property  which  is  the  cause  of  motion  pro 
duced,  changed,  or  prevented.  This  conception  is,  it 
would  seem,  mainly  and  primarily  suggested  by  our 
consciousness  of  the  exertions  by  which  we  put  bodies 
in  motion.  The  Latin  and  Greek  words  for  Force,  Vis, 
FJV,  were  probably,  like  all  abstract  terms,  derived  at 
first  from  some  sensible  object.  The  original  meaning 
of  the  Greek  word  was  a  muscle  or  tendon.  Its  first 
application  as  an  abstract  term  is  accordingly  to  muscu 
lar  force. 

aJr'   Amc   iro\v  /ueioi/«   Aaai/  «ei^«<; 
rfK    t*7riC(i/>;'(Tac,   eTrepeifff    (e   FIN    direXfflpov. 

Then  Ajax  a  far  heavier  stone  upheaved, 
He  whirled  it,  and  impressing  Force  intense 
Upon  the  mass,  dismist  it. 

The  property  by   which   bodies  affect  each  other's 
motions,  was  naturally  likened  to  that  energy  which  we 


exert  upon  them  with  similar  effect :  and  thus  the  labour 
ing  horse,  the  rushing  torrent,  the  descending  weight,  the 
elastic  bow,  were  said  to  exert  force.  Homer*  speaks 
of  the  force  of  the  river,  R'?  Trorano'io ;  and  Hesiodf  of 
the >force  of  the  north  wind,  F<s  dve^ov  fiopeao. 

Thus  man's  general  notion  of  force  was  probably  first 
suggested  by  his  muscular  exertions,  that  is,  by  an  act 
depending  upon  that  muscular  sense,  to  which,  as  we 
have  already  seen,  the  perception  of  space  is  mainly  due. 
And  this  being  the  case,  it  will  be  easily  understood  that 
the  Direction  of  the  force  thus  exerted  is  perceived  by 
the  muscular  sense,  at  the  same  time  that  the  force  itself 
is  perceived ;  and  that  the  direction  of  any  other  force  is 
understood  by  comparison  with  force  which  man  must 
exert  to  produce  the  same  effect,  in  the  same  manner  as 
force  itself  is  so  understood. 

This  abstract  notion  of  Force  long  remained  in  a  very 
vague  and  obscure  condition,  as  may  be  seen  by  referring 
to  the  History  for  the  failures  of  attempts  at  a  science  of 
force  and  motion,  made  by  the  ancients  and  their  com 
mentators  in  the  middle  ages.  By  degrees,  in  modern 
times,  we  see  the  scientific  faculty  revive.  The  concep 
tion  of  Force  becomes  so  far  distinct  and  precise  that  it 
can  be  reasoned  upon  in  a  consistent  manner,  with  de 
monstrated  consequences  ;  and  a  genuine  science  of  Me 
chanics  conies  into  existence.  The  foundations  of  this 
science  are  to  be  found  in  the  Axioms  concerning  causa 
tion  which  we  have  already  stated ;  these  axioms  being 
interpreted  and  fixed  in  their  application  by  a  constant 
reference  to  observed  facts,  as  we  shall  show.  But  we 
must,  in  the  first  place,  consider  further  those  primary 
processes  of  observation  by  which  we  acquire  the  first 
materials  of  thought  on  such  subjects. 

2.  Matter. — The  conception  of  Force,  as  we  have  said, 
*  //.  xxr.  t  Op.  ct  D. 


arises  with  our  consciousness  of  our  own  muscular  exer 
tions.  But  we  cannot  imagine  such  exertions  without 
also  imagining  some  bodily  substance  against  which  they 
are  exercised.  If  we  press,  we  press  something :  if  we 
thrust  or  throw,  there  must  be  something  to  resist  the 
thrust  or  to  receive  the  impulse.  Without  body,  mus 
cular  force  cannot  be  exerted  and  force  in  general  is  not 

Thus  Force  cannot  exist  without  Body  on  which  it 
acts.  The  two  conceptions,  Force  and  Matter,  are  co 
existent  and  correlative.  Force  implies  resistance ;  and 
the  force  is  effective  only  wheiL  the  resistance  Js  called 
into  j)lay_.  If  we  grasp  a  stone,  we  have  no  hold  of  it 
till  the  closing  of  the  hand  is  resisted  by  the  solid  tex 
ture  of  the  stone.  If  we  push  open  a  gate,  we  must 
surmount  the  opposition  which  it  exerts  while  turning 
on  its  hinges.  However  slight  the  resistance  be,  there 
must  be  some  resistance,  or  there  would  be  no  force. 
If  we  imagine  a  state  of  things  in  which  objects  do  not 
resist  our  touch,  they  must  also  cease  to  be  influenced 
by  our  strength.  Such  a  state  of  things  we  sometimes 
imagine  in  our  dreams;  and  such  are  the  poetical  pic 
tures  of  the  regions  inhabited  by  disembodied  spirits.  In 
these,  the  figures  which  appear  are  conspicuous  to  the 
eye,  but  impalpable  like  shadow  or  smoke ;  and  as  they 
do  not  resist  the  corporeal  impressions,  so  neither  do 
they  obey  them.  The  spectator  tries  in  vain  to  strike 
or  to  grasp  them. 

Et  ni  cana  vates  tcnues  sine  corpore  vitas 
A  dm  volitare  c.iva  sub  imagine  forma1, 
Irruat  ac  frustra  ferro  diverbcret  umbras. 

The  Sibyl  warns  him  that  there  round  him  fly 
Bodiless  things,  but  substance  to  the  eye ; 
Else  had  he  pierced  those  shapes  with  life-like  face, 
And  smitten,  fierce,  the  unresisting  space. 


Neque  ilium 

Prensantem  nequicquam  umbras  et  multa  volentem 
Diccrc,  preterca  vidit. 
He  grasps  her  form,  and  clutches  but  the  shade. 

Such  may  be  the  circumstances  of  the  unreal  world  of 
dreams,  or  of  poetical  fancies  approaching  to  dreams : 
for  in  these  worlds  our  imaginary  perceptions  are  bound 
by  no  rigid  conditions  of  force  and  reaction.  In  such 
cases,  the  mind  casts  off  the  empire  of  the  idea  of  cause, 
as  it  casts  off  even  the  still  more  familiar  sway  of  the 
ideas  of  space  and  time.  But  the  character  of  the 
material  world  in  which  we  live  when  awake  is,  that  we 
have  at  every  instant  and  at  every  place,  force  operating 
on  matter  and  matter  resisting  force. 

3.  Solidity. — From  our  consciousness  of  muscular 
exertion,  we  derive,  as  we  have  seen,  the  conception  of 
force,  and  with  that  also  the  conception  of  matter.  We 
have  already  shown,  in  a  former  chapter,  that  the  same 
part  of  our  frame,  the  muscular  system,  is  the  organ  by 
which  we  perceive  extension  and  the  relations  of  space. 
Thus  the  same  organ  gives  us  the  perception  of  body  as 
resisting  force,  and  as  occupying  space ;  and  by  combin 
ing  these  conditions  we  have  the  conception  of  solid 
extended  bodies.  In  reality,  this  resistance  is  inevitably 
presented  to  our  notice  in  the  very  facts  from  which  we 
collect  the  notion  of  extension.  For  the  action  of  the 
hand  and  arm  by  which  we  follow  the  forms  of  objects, 
implies  that  we  apply  our  fingers  to  their  surface ;  and 
we  are  stopped  there  by  the  resistance  which  the  body 
offers.  This  resistance  is  precisely  that  which  is  requisite 
in  order  to  make  us  conscious  of  our  muscular  effort*. 
Neither  touch,  nor  any  other  mere  passive  sensation, 
could  produce  the  perception  of  extent,  as  we  have 
already  urged  :  nor  could  the  muscular  sense  lead  to  such 

*  Brown's  Lectures,  i.  46fi. 


a  perception,  except  the  extension  of  the  muscles  were 
felt  to  be  resisted.  And  thus  the  perception  of  resistance 
enters  the  mind  along  with  the  perception  of  extended 
bodies.  All  the  objects  with  which  we  have  to  do  are 
not  only  extended  but  solid. 

This  sense  of  the  term  solidity,  (the  general  property 
of  all  matter,)  is  different  to  that  in  which  we  oppose 
stolidity  to  Jiuidity.  We  may  avoid  ambiguity  by  op 
posing  rigid  to  fluid  bodies.  By  solid  bodies,  as  we  now 
speak  of  them,  we  mean  only  such  as  resist  the  pressure 
which  we  exert,  so  long  as  their  parts  continue  in  their 
places.  By  fluid  bodies,  we  mean  those  whose  parts  are, 
by  a  slight  pressure,  removed  out  of  their  places.  A  drop 
of  water  ceases  to  prevent  the  contact  of  our  two  hands, 
not  by  ceasing  to  have  solidity  in  this  sense,  but  by  being 
thrust  out  of  the  way.  If  it  could  remain  in  its  place, 
it  could  not  cease  to  exercise  its  resistance  to  our  pres 
sure,  except  by  ceasing  to  be  matter  altogether. 

The  perception  of  solidity,  like  the  perception  of 
extension,  implies  an  act  of  the  mind,  as  well  as  an 
impression  of  the  senses :  as  the  perception  of  extension 
implies  the  idea  of  space,  so  the  perception  of  solidity 
implies  the  idea  of  action  and  reaction.  That  an  Idea 
is  involved  in  our  knowledge  on  this  subject  appears,  as 
in  other  instances,  from  this  consideration,  that  the  con 
victions  of  persons,  even  of  those  who  allow  of  no  ground 
of  knowledge  but  experience,  do  in  fact  go  far  beyond  the 
possible  limits  of  experience.  Thus  Locke  says*,  that 
"  the  bodies  which  we  daily  handle  hinder  by  an  insur 
mountable  force  the  approach  of  the  parts  of  our  hands 
that  press  them."  Now  it  is  manifest  that  our  observa 
tion  can  never  go  to  this  length.  By  our  senses  we  can 
only  perceive  that  bodies  resist  the  greatest  actual  forces 
that  we  exert  upon  them.  But  our  conception  of  force 

*  E&say,  B.  ii.  c.  4. 


carries  us  further :  and  since,  so  long  as  the  body  is 
there  to  receive  the  action  of  the  force,  it  must  suffer 
the  whole  of  that  action,  and  must  react  as  much  as 
it  suffers :  it  is  therefore  true,  that  so  long  as  the  body 
remains  there,  the  force  which  is  exerted  upon  it  can 
never  surmount  the  resistance  which  the  body  exercises. 
And  thus  this  doctrine,  that  bodies  resist  the  intrusion 
of  other  bodies  by  an  insurmountable  force,  is,  in  fact, 
a  consequence  of  the  axiom  that  the  reaction  is  always 
equal  to  the  action. 

4.  Inertia. — But  this  principle  of  the  equality  of 
action  and  reaction  appears  also  in  another  way.  Not 
only  when  we  exert  force  upon  bodies  at  rest,  but  when, 
by  our  exertions,  we  put  them  in  motion,  they  react.  If 
we  set  a  large  stone  in  motion,  the  stone  resists ;  for  the 
operation  requires  an  effort.  By  increasing  the  effort,  we 
can  increase  the  effect,  that  is,  the  motion  produced ;  but 
the  resistance  still  remains.  And  the  greater  the  stone 
moved,  the  greater  is  the  effort  requisite  to  move  it. 
There  is,  in  every  case,  a  resistance  to  motion,  which  shows 
itself,  not  in  preventing  the  motion,  but  in  a  reciprocal 
force,  exerted  backwards  upon  the  agent  by  which  the 
motion  is  produced.  And  this  resistance  resides  in 
each  portion  of  matter,  for  it  is  increased  as  we  add 
one  portion  of  matter  to  another.  We  can  push  a  light 
boat  rapidly  through  the  water ;  but  we  may  go  on 
increasing  its  freight,  till  we  are  barely  able  to  stir  it. 
This  property  of  matter,  then,  by  which  it  resists  the 
reception  of  motion,  or  rather  by  which  it  reacts  and 
requires  an  adequate  force  in  order  that  any  motion  may 
result,  is  called  its  inertness,  or  inertia.  That  matter  has 
such  a  property,  is  a  conviction  flowing  from  that  idea  of 
a  reaction  equal  and  opposite  to  the  action,  which  the 
conception  of  all  force  involves.  By  what  laws  this 
inertia  depends  on  the  magnitude,  form,  and  material  of 


the  body,  must  be  the  subject  of  our  consideration  here 
after.  But  that  matter  has  this  inertia,  in  virtue  of 
which,  as  the  matter  is  greater,  the  velocity  which  the 
same  effort  can  communicate  to  it  is  less,  is  a  principle 
inseparable  from  the  notion  of  matter  itself. 

Hermann  says  that  Kepler  first  introduced  this  "  most 
significant  word"  inertia.  Whether  it  is  to  be  found  in 
earlier  writers  I  know  not ;  Kepler  certainly  does  use  it 
familiarly  in  those  attempts  to  assign  physical  reasons 
for  the  motions  of  the  planets  which  were  among  the 
main  occasions  of  the  discovery  of  the  true  laws  of  me 
chanics.  He  assumes  the  slowness  of  the  motions  of  the 
planets  to  increase,  (other  causes  remaining  the  same,) 
as  the  inertia  increases ;  and  though,  even  in  this  as 
sumption,  there  is  an  errour  involved,  (if  we  adopt  that 
interpretation  of  the  term  inertia  to  which  subsequent 
researches  led,)  the  introduction  of  such  a  word  was  one 
step  in  determining  and  expressing  those  laws  of  motion 
which  depend  on  the  fundamental  principle  of  the  equality 
of  action  and  reaction. 

5.  We  have  thus  seen,  I  trust  in  a  satisfactory 
manner,  the  origin  of  our  conceptions  of  Force,  Matter, 
Solidity,  and  Inertness.  It  has  appeared  that  the  organ 
by  which  we  obtain  such  conceptions  is  that  very  mus 
cular  frame,  which  is  the  main  instrument  of  our  percep 
tions  of  space ;  but  that,  besides  bodily  sensations,  these 
ideal  conceptions,  like  all  the  others  which  we  have 
hitherto  considered,  involve  also  an  habitual  activity  of 
the  mind,  giving  to  our  sensations  a  meaning  which  they 
could  not  otherwise  possess.  And  among  the  ideas  thus 
brought  into  play,  is  an  idea  of  action  with  an  equal  and 
opposite  reaction,  which  forms  a  foundation  for  univer 
sal  truths  to  be  hereafter  established  respecting  the 
conceptions  thus  obtained. 

We  must  now  endeavour  to  trace  in  what  manner 


these  fundamental  principles  and  conceptions  are  un 
folded  by  means  of  observation  and  reasoning,  till  they 
become  an  extensive  yet  indisputable  science. 



1.  Object  of  the  Chapter. — IN  the  present  and  the 
succeeding  chapters  we  have  to  show  how  the  general 
axioms  of  Causation  enable  us  to  construct  the  science 
of  Mechanics.  We  have  to  consider  these  axioms  as 
moulding  themselves,  in  the  first  place,  into  certain  fun 
damental  mechanical  principles,  which  are  of  evident 
and  necessary  truth  in  virtue  of  their  dependence  upon 
the  general  axioms  of  Causation  ;  and  thus  as  forming  a 
foundation  for  the  whole  structure  of  the  science ; — a 
system  of  truths  no  less  necessary  than  the  fundamen 
tal  principles,  because  derived  from  these  by  rigorous 

This  account  of  the  construction  of  the  science  of 
Mechanics,  however  generally  treated,  cannot  be  other 
wise  than  technical  in  its  details,  and  will  probably  be 
imperfectly  understood  by  any  one  not  acquainted  with 
Mechanics  as  a  mathematical  science. 

I  cannot  omit  this  portion  of  my  survey  without 
rendering  my  work  incomplete ;  but  I  may  remark  that 
the  main  purpose  of  it  is  to  prove,  in  a  more  particular 
manner,  what  I  have  already  declared  in  general,  that 
there  are,  in  Mechanics  no  less  than  in  Geometry,  funda 
mental  principles  of  axiomatic  evidence  and  necessity; 
— that  these  principles  derive  their  axiomatic  character 
from  the  Idea  which  they  involve,  namely  the  Idea  of 


Cause ; — and  that  through  the  combination  of  principles 
of  this  kind,  the  whole  science  of  Mechanics,  including 
its  most  complex  and  remote  results,  exists  as  a  body  of 
solid  and  universal  truths. 

2.  Statics  and  Dynamics. — We  must  first  turn  our 
attention  to  a  technical  distinction  of  Mechanics   into 
two  portions,  according  as  the  forces  about  which  we 
reason  produce  rest,  or  motion ;   the  former  portion  is 
termed   Statics,  the  latter  Dynamics.     If  a  stone  fall, 
or   a   weight   put  a  machine    in    motion,    the   problem 
belongs  to   Dynamics;    but  if  the  stone  rest  upon  the 
ground,  or  a  weight  be  merely  supported  by  a  machine, 
without  being   raised    higher,    the   question    is    one    of 

3.  Equilibrium. — In   Statics,    forces    balance    each 
other,  or  keep  each  other  in  equilibrium.     And  forces 
which  directly  balance  each  other,  or  keep  each  other  in 
equilibrium,   are    necessarily  and   manifestly  equal.     If 
we  see  two  boys   pull  at  two  ends  of  a  rope  so   that 
neither  of  them  in  the  smallest  degree  prevails  over  the 
other,  we  have  a  case  in  which  two  forces  are  in  equili 
brium.     The  two  forces  are  evidently  equal,  and  are  a 
statical  exemplification  of  action  and  reaction,  such  as  are 
spoken  of  in  the  third  axiom  concerning  causes.     Now 
the  same  exemplification  occurs  in  every  case  of  equili 
brium.     No  point  or  body  can  be  kept  at  rest  except  in 
virtue  of  opposing  forces  acting  upon  it ;  and  these  forces 
must  always  be  equal  in  their  opposite  effect.     When  a 
stone  lies  on  the  floor,  the  weight  of  the  stone  down 
wards  is  opposed  and  balanced  by  an  equal  pressure  of 
the  floor  upwards.     If  the  stone  rests  on  a    slope,   its 
tendency  to    slide  is  counteracted    by  some  equal   and 
opposite    force,  arising,  it  may  be,  from  the  resistance 
which  the  sloping  ground  opposes  to  any  motion  along 
its  surface.     Every  case  of  rest  is  a  case  of  equilibrium  : 

VOL.  i.    \v.  P.  0 


every  case  of  equilibrium  is  a  case  of  equal  and  opposite 

The  most  complex  frame-work  on  which  weights  are 
supported,  as  the  roof  of  a  building,  or  the  cordage  of  a 
machine,  are  still  examples  of  equilibrium.  In  such 
cases  we  may  have  many  forces  all  combining  to  balance 
each  other ;  and  the  equilibrium  will  depend  on  various 
conditions  of  direction  and  magnitude  among  the  forces. 
And  in  order  to  understand  what  are  these  conditions, 
we  must  ask,  in  the  first  place,  what  we  understand  by 
the  magnitude  of  such  forces ; — what  is  the  measure  of 
statical  forces. 

4.  Measure  of  Statical  Forces. — At  first  we  might 
expect,  perhaps,  that  since  statical  forces  come  under  the 
general  notion  of  Cause,  the  mode  of  measuring  them 
would  be  derived  from  the  second  axiom  of  Causation, 
that  causes  are  measured  by  their  effects.  But  we  find 
that  the  application  of  this  axiom  is  controlled  by  the 
limitation  which  we  noticed,  after  stating  that  axiom ; 
namely,  the  condition  that  the  causes  shall  be  capable  of 
addition.  Further,  as  we  have  seen,  a  statical  force  pro 
duces  no  other  effect  than  this,  that  it  balances  some 
other  statical  force ;  and  hence  the  measure  of  statical 
forces  is  necessarily  dependent  upon  their  balancing, 
that  is,  upon  the  equality  of  action  and  reaction. 

That  statical  forces  are  capable  of  addition  is  involved 
in  our  conception  of  such  forces.  When  two  men  pull 
at  a  rope  in  the  same  direction,  the  forces  which  they 
exert  are  added  together.  When  two  heavy  bodies  are 
put  into  a  basket  suspended  by  a  string,  their  weights 
are  added,  and  the  sum  is  supported  by  the  string. 

Combining  these  considerations,  it  will  appear  that 
the  measure  of  statical  forces  is  necessarily  given  at  once 
by  the  fundamental  principle  of  the  equality  of  action 
and  reaction.  Since  two  opposite  forces  which  balance 


oach  other  are  equal,  each  force  is  measured  by  that 
which  it  balances ;  and  since  forces  are  capable  of  addi 
tion,  a  force  of  any  magnitude  is  measured  by  adding  to 
gether  a  proper  number  of  such  equal  forces.  Thus  a 
heavy  body  which,  appended  to  some  certain  elastic 
branch  of  a  tree,  would  bend  it  down  through  one  inch, 
•may  be  taken  as  a  unit  of  weight.  Then  if  we  remove 
this  first  body,  and  find  a  second  heavy  body  which  will 
also  bend  the  branch  through  the  same  space,  this  is  also 
a  unit  of  weight ;  and  in  like  manner  we  might  go  on  to 
a  third  and  a  fourth  equal  body ;  and  adding  together 
the  two,  or  the  three,  or  the  four  heavy  bodies,  we  have 
a  force  twice,  or  three  times,  or  four  times  the  unit  of 
•weight.  And  with  such  a  collection  of  heavy  bodies,  or 
weights,  we  can  readily  measure  all  other  forces ;  for  the 
same  principle  of  the  equality  of  action  and  reaction 
leads  at  once  to  this  maxim,  that  any  statical  force  is 
measured  by  the  weight  which  it  would  support. 

As  has  been  said,  it  might  at  first  have  been  sup 
posed  that  we  should  have  to  apply,  in  this  case,  the 
axiom  that  causes  are  measured  by  their  effects  in  an 
other  manner;  that  thus,  if  that  body  were  a  unit  of 
weight  which  bent  the  bough  of  a  tree  through  one  inch, 
that  body  would  be  two  units  which  bent  it  through  two 
inches,  and  so  on.  But,  as  we  have  already  stated,  the 
measures  of  weight  must  be  subject  to  this  condition, 
that  they  are  susceptible  of  being  added :  and  therefore 
we  cannot  take  the  deflexion  of  the  bough  for  our  mea 
sure,  till  we  have  ascertained,  that  which  experience 
alone  can  teach  us,  that  under  the  burden  of  two  equal 
weights,  the  deflexion  will  be  twice  as  great  as  it  is  with 
one  weight,  which  is  not  true,  or  at  least  is  neither  ob 
viously  nor  necessarily  true.  In  this,  as  in  all  other  cases, 
although  causes  must  be  measured  by  their  effects,  we 
learn  from  experience  only  how  the  effects  are  to  be 


interpreted,  so  as  to  give  a  true  and  consistent  mea 

With  regard,  however,  to  the  measure  of  statical 
force,  and  of  weight,  no  difficulty  really  occurred  to  phi 
losophers  from  the  time  when  they  first  began  to  specu 
late  on  such  subjects ;  for  it  was  easily  seen  that  if  we 
take  any  uniform  material,  as  wood,  or  stone,  or  iron, 
portions  of  this  which  are  geometrically  equal,  must  also 
be  equal  in  statical  effect ;  since  this  was  implied  in  the 
very  hypothesis  of  a  uniform  material.  And  a  body  ten 
times  as  large  as  another  of  the  same  substance,  will  be 
of  ten  times  the  weight.  But  before  men  could  esta 
blish  by  reasoning  the  conditions  under  which  weights 
would  be  in  equilibrium,  some  other  principles  were 
needed  in  addition  to  the  mere  measure  of  forces.  The 
principles  introduced  for  this  purpose  still  resulted  from 
the  conception  of  equal  action  and  reaction ;  but  it  re 
quired  no  small  clearness  of  thought  to  select  them 
rightly,  and  to  employ  them  successfully.  This,  however, 
was  done,  to  a  certain  extent,  by  the  Greeks ;  and  the 
treatise  of  Archimedes  On  the  Center  of  Gravity,  is 
founded  on  principles  which  may  still  be  considered  as 
the  genuine  basis  of  statical  reasoning.  I  shall  make  a 
few  remarks  on  the  most  important  principle  among 
those  which  Archimedes  thus  employs. 

5.  The  Center  of  Gravity. — The  most  important  of 
the  principles  which  enter  into  the  demonstration  of 
Archimedes  is  this :  that  "  Every  body  has  a  center  of 
gravity ;"  meaning  by  the  center  of  gravity,  a  point  at 
which  the  whole  matter  of  the  body  may  be  supposed  to 
be  collected,  to  all  intents  and  purposes  of  statical 
reasoning.  This  principle  has  been  put  in  various  forms 
by  succeeding  writers :  for  instance,  it  has  been  thought 
sufficient  to  assume  a  case  much  simpler  than  the  general 
one;  and  to  assert  that  two  equal  bodies  have  their 


center  of  gravity  in  the  point  midway  between  them.  It 
is  to  be  observed,  that  this  assertion  not  only  implies 
that  the  two  bodies  will  balance  upon  a  support  placed 
at  that  midway  point,  but  also,  that  they  will  exercise, 
upon  such  a  support,  a  pressure  equal  to  their  sum ; 
for  this  point  being  the  center  of  gravity,  the  whole 
matter  of  the  two  bodies  may  be  conceived  to  be  col 
lected  there,  and  therefore  the  whole  weight  will  press 
there.  And  thus  the  principle  in  question  amounts  to 
this,  that  when  two  equal  heavy  bodies  are  supported  on 
the  middle  point  between  them,  the  pressure  upon  the 
support  is  equal  to  the  sum  of  the  weights  of  the  bodies. 

A  clear  understanding  of  the  nature  and  grounds  of 
this  principle  is  of  great  consequence :  for  in  it  we  have 
the  foundation  of  a  large  portion  of  the  science  of 
Mechanics.  And  if  this  principle  can  be  shown  to  be 
necessarily  true,  in  virtue  of  our  Fundamental  Ideas,  we 
can  hardly  doubt  that  there  exist  many  other  truths  of 
the  same  kind,  and  that  no  sound  view  of  the  evidence 
and  extent  of  human  knowledge  can  be  obtained,  so  long 
as  we  mistake  the  nature  of  these,  its  first  principles. 

The  above  principle,  that  the  pressure  on  the  support 
is  equal  to  the  sum  of  the  bodies  supported,  is  often 
stated  as  an  axiom  in  the  outset  of  books  on  Mechanics. 
And  this  appears  to  be  the  true  place  and  character  of 
this  principle,  in  accordance  with  the  reasonings  which 
we  have  already  urged.  The  axiom  depends  upon  our 
conception  of  action  and  reaction.  That  the  two  weights 
are  supported,  implies  that  the  supporting  force  must  be 
equal  to  the  force  or  weight  supported. 

In  order  further  to  show  the  foundation  of  this 
principle,  we  may  ask  the  question : — If  it  be  not  an 
axiom,  deriving  its  truth  from  the  fundamental  concep 
tion  of  equal  action  and  reaction,  which  equilibrium 
always  implies,  what  is  the  origin  of  its  certainty?  The 


principle  is  never  for  an  instant  denied  or  questioned:  it  is 
taken  for  granted,  even  before  it  is  stated.  No  one  will 
doubt  that  it  is  not  only  true,  but  true  with  the  same 
rigour  and  universality  as  the  axioms  of  Geometry.  Will 
it  be  said,  that  it  is  borrowed  from  experience  ?  Expe 
rience  could  never  prove  a  principle  to  be  universally 
and  rigorously  true.  Moreover,  when  from  experience 
we  prove  a  proposition  to  possess  great  exactness  and 
generality,  we  approach  by  degrees  to  this  proof:  the 
conviction  becomes  stronger,  the  truth  more  secure,  as 
we  accumulate  trials.  But  nothing  of  this  kind  is  the 
case  in  the  instance  before  us.  There  is  no  gradation 
from  less  to  greater  certainty ; — no  hesitation  which 
precedes  confidence.  From  the  first,  we  know  that  the 
axiom  is  exactly  and  certainly  true.  In  order  to  be 
convinced  of  it,  we  do  not  require  many  trials,  but 
merely  a  clear  understanding  of  the  assertion  itself. 

But  in  fact,  not  only  are  trials  not  necessary  to  the 
proof,  but  they  do  not  strengthen  it.  Probably  no 
one  ever  made  a  trial  for  the  purpose  of  showing  that 
the  pressure  upon  the  support  is  equal  to  the  sum  of  the 
two  weights.  Certainly  no  person  with  clear  mechanical 
conceptions  ever  wanted  such  a  trial  to  convince  him  of 
the  truth ;  or  thought  the  truth  clearer  after  the  trial 
had  been  made.  If  to  such  a  person,  an  experiment 
were  shown  which  seemed  to  contradict  the  principle,  his 
conclusion  would  be,  not  that  the  principle  was  doubtful, 
but  that  the  apparatus  was  out  of  order.  Nothing  can 
be  less  like  collecting  truth  from  experience  than  this. 

We  maintain,  then,  that  this  equality  of  mechanical 
action  and  reaction,  is  one  of  the  principles  which  do 
not  flow  from,  but  regulate  our  experience.  To  this 
principle,  the  facts  which  we  observe  must  conform ; 
and  we  cannot  help  interpreting  them  in  such  a  manner 
that  they  shall  be  exemplifications  of  the  principle.  A 


mechanical  pressure  not  accompanied  by  an  equal  and 
opposite  pressure,  can  no  more  be  given  by  experience, 
than  two  unequal  right  angles.  With  the  supposition  of 
such  inequalities,  space  ceases  to  be  space,  force  ceases  to 
be  force,  matter  ceases  to  be  matter.  And  this  equality 
of  action  and  reaction,  considered  in  the  case  in  which 
two  bodies  are  connected  so  as  to  act  on  a  single  support, 
leads  to  the  axiom  which  we  have  stated  above,  and 
which  is  one  of  the  main  foundations  of  the  science  of 

6.  Oblique  Forces. — By  the  aid  of  this  axiom  and 
a  few  others,  the  Greeks  made  some  progress  in  the 
science  of  Statics.  But  after  a  short  advance,  they 
arrived  at  another  difficulty,  that  of  Oblique  Forces, 
which  they  never  overcame;  and  which  no  mathematician 
mastered  till  modern  times.  The  unpublished  manuscripts 
of  Leonardo  da  Vinci,  written  in  the  fifteenth  century, 
and  the  works  of  Stevinus  and  Galileo,  in  the  sixteenth, 
are  the  places  in  which  we  find  the  first  solid  grounds  of 
reasoning  on  the  subject  of  forces  acting  obliquely  to 
each  other.  And  mathematicians,  having  thus  become 
possessed  of  all  the  mechanical  principles  which  are 
requisite  in  problems  respecting  equilibrium,  soon  framed 
a  complete  science  of  Statics.  Succeeding  writers  pre 
sented  this  science  in  forms  variously  modified ;  for  it 
was  found,  in  Mechanics  as  in  Geometry,  that  various 
propositions  might  be  taken  as  the  starting  points ;  and 
that  the  collection  of  truths  which  it  was  the  mecha 
nician's  business  to  include  in  his  course,  might  thus  be 
traversed  by  various  routes,  each  path  offering  a  series 
of  satisfactory  demonstrations.  The  fundamental  con 
ceptions  of  force  and  resistance,  like  those  of  space  and 
number,  could  be  contemplated  under  different  aspects, 
each  of  which  might  be  made  the  basis  of  axioms, 
or  of  principles  employed  as  axioms.  Hence  the 


grounds  of  the  truth  of  Statics  may  be  stated  in  various 
ways ;  and  it  would  be  a  task  of  some  length  to  examine 
all  these  completely,  and  to  trace  them  to  their  Funda 
mental  Ideas.  This  I  shall  not  undertake  here  to  do ; 
but  the  philosophical  importance  of  the  subject  makes 
it  proper  to  offer  a  few  remarks  on  some  of  the  main 
principles  involved  in  the  different  modes  of  presenting 
Statics  as  a  rigorously  demonstrated  science. 

7.  A  Force  may  be  supposed  to  act  at  any  Point  of  its 
Direction. — It  has  been  stated  in  the  history  of  Mecha 
nics*,  that  Leonardo  da  Vinci  and  Galileo  obtained  the 
true  measure  of  the  effect  of  oblique  forces,  by  reason 
ings  which  were,  in  substance,  the  same.  The  principle 
of  these  reasonings  is  that  expressed  at  the  head  of  this 
paragraph ;  and  when  we  have  a  little  accustomed  our 
selves  to  contemplate  our  conceptions  of  force,  and  its 
action  on  matter,  in  an  abstract  manner,  we  shall  have 
no  difficulty  in  assenting  to  the  principle  in  this  general 
form.  But  it  may,  perhaps,  be  more  obvious  at  first  in 
a  special  case. 

If  we  suppose  a  wheel,  moveable  about  its  axis,  and 
carrying  with  it  in  its  motion  a  weight,  (as,  for  example, 
one  of  the  wheels  by  means  of  which  the  large  bells  of  a 
church  are  rung,)  this  weight  may  be  supported  by  means 
of  a  rope  (not  passing  along  the  circumference  of  the 
wheel,  as  is  usual  in  the  case  of  bells,)  but  fastened  to 
one  of  the  spokes  of  the  wheel.  Now  the  principle  which 
is  enunciated  above  asserts,  that  if  the  rope  pass  in  a 
straight  line  across  several  of  the  spokes  of  the  wheel,  it 
makes  no  difference  in  the  mechanical  effect  of  the  force 
applied,  for  the  purpose  of  putting  the  bell  in  motion,  to 
which  of  these  spokes  the  rope  is  fastened.  In  each  case, 
the  fastening  of  the  rope  to  the  wheel  merely  serves  to 
enable  the  force  to  produce  motion  about  the  centre ; 

*  Hist.  Ind.  ScL,  B.  vi.  c.  i.  sect.  2.  and  Note  (A). 


and  so  long  as  the  force  acts  in  the  same  line,  the  effect 
is  the  same,  at  whatever  point  of  the  rope  the  line  of 
action  finishes. 

This  axiom  very  readily  aids  us  in  estimating  the 
effect  of  oblique  forces.  For  when  a  force  acts  on  one  of 
the  arms  of  a  lever  at  any  oblique  angle,  we  suppose 
another  arm  projecting  from  the  centre  of  motion,  like 
another  spoke  of  the  same  wheel,  so  situated  that  it  is 
perpendicular  to  the  force.  This  arm  we  may,  with 
Leonardo,  call  the  virtual  lexer ;  for,  by  the  axiom,  we 
may  suppose  the  force  to  act  where  the  line  of  its  direc 
tion  meets  this  arm ;  and  thus  we  reduce  the  case  to 
that  in  which  the  force  acts  perpendicularly  on  the  arm. 
The  ground  of  this  axiom  is,  that  matter,  in  Statics, 
is  necessarily  conceived  as  transmitting  force.  That  force 
can  be  transmitted  from  one  place  to  another,  by  means 
of  matter ; — that  we  can  push  with  a  rod,  pull  with  a 
rope, — are  suppositions  implied  in  our  conceptions  of 
force  and  matter.  Matter  is,  as  we  have  said,  that  which 
receives  the  impression  of  force,  and  the  modes  just 
mentioned,  are  the  simplest  ways  in  which  that  impres 
sion  operates.  And  since,  in  any  of  these  cases,  the  force 
might  be  resisted  by  a  reaction  equal  to  the  force  itself, 
the  reaction  in  each  case  would  be  equal,  and,  therefore, 
the  action  in  each  case  is  necessarily  equal ;  and  thus  the 
forces  must  be  transmitted,  from  one  point  to  another, 
without  increase  or  diminution. 

This  property  of  matter,  of  transmitting  the  action  of 
force,  is  of  various  kinds.  We  have  the  coherence  of  a 
rope  which  enables  us  to  pull,  and  the  rigidity  of  a  staftj 
which  enables  us  to  push  with  it  in  the  direction  of  its 
length ;  and  again,  the  same  staff  has  a  rigidity  of  another 
kind,  in  virtue  of  which  we  can  use  it  as  a  lever ;  that  is,  a 
rigidity  to  resist  flexure,  and  to  transmit  the  force  which 
turns  a  body  round  a  fulcrum.  There  is,  further,  the 


rigidity  by  which  a  solid  body  resists  twisting.  Of  these 
kinds  of  rigidity,  the  first  is  that  to  which  our  axiom 
refers ;  but  in  order  to  complete  the  list  of  the  ele 
mentary  principles  of  Statics,  we  ought  also  to  lay  down 
axioms  respecting  the  other  kinds  of  rigidity*.  These, 
however,  I  shall  not  here  state,  as  they  do  not  involve 
any  new  principle.  Like  the  one  just  considered,  they 
form  part  of  our  fundamental  conception  of  matter  ;  they 
are  not  the  results  of  any  experience,  but  are  the  hypo 
theses  to  which  we  are  irresistibly  led,  when  we  would 
liberate  our  reasonings  concerning  force  and  matter  from 
a  dependence  on  the  special  results  of  experience.  We 
cannot  even  conceive  (that  is,  if  we  have  any  clear 
mechanical  conceptions  at  all)  the  force  exerted  by  the 
point  of  a  staff  and  resisting  the  force  which  we  steadily 
impress  on  the  head  of  it,  to  be  different  from  the 
impressed  force. 

8.  Forces  may  have  equivalent  Forces  substituted  for 
them.  The  Parallelogram  of  Forces. — It  has  already  been 
observed,  that  in  order  to  prove  the  doctrines  of  Statics, 
we  may  take  various  principles  as  our  starting  points, 
and  may  still  find  a  course  of  demonstration  by  which 
the  leading  propositions  belonging  to  the  subject  may 
be  established.  Thus,  instead  of  beginning  our  reason 
ings,  as  in  the  last  section  we  supposed  them  to 
commence,  with  the  case  in  which  forces  act  upon 
different  points  of  the  same  body  in  the  same  line  of 
force,  and  counteract  each  other  in  virtue  of  the  inter 
vening  matter  by  which  the  effect  of  force  is  transferred 
from  one  point  to  another,  we  may  suppose  different 
forces  to  act  at  the  same  point,  and  may  thus  commence 
our  reasonings  with  a  case  in  which  we  have  to  con 
template  force,  without  having  to  take  into  our  account 

*  Such  axioms  are  given  in  a  little  work  (The  Mechanical  Euclid} 
which  I  published  on  the  Elements  of  Mechanics. 


the  resistance  or  rigidity  of  matter.  Two  statical  forces, 
thus  acting  at  a  mathematical  point,  are  equivalent,  in 
al  1  respects,  to  some  single  force  acting  at  tjie  same  point ; 
and  would  be  kept  in  equilibrium  by  a  force  equal  and 
opposite  to  that  single  force.  And  the  rule  by  which 
the  single  force  is  derived  from  the  two,  is  commonly 
termed  the  parallelogram  offerees;  the  proposition  being 
this, — That  if  the  two  forces  be  represented  in  magnitude 
and  direction  by  the  two  sides  of  a  parallelogram,  the 
resulting  force  will  be  represented  in  the  same  manner 
by  the  diagonal  of  the  parallelogram.  This  proposition 
has  very  frequently  been  made,  by  modern  writers,  the 
commencement  of  the  science  of  Mechanics :  a  position 
for  which,  by  its  simplicity,  it  is  well  suited ;  although, 
in  order  to  deduce  from  it  the  other  elementary  proposi 
tions  of  the  science,  as,  for  instance,  those  respecting  the 
lever,  we  require  the  axiom  stated  in  the  last  section. 

9.  The  Parallelogram  of  Forces  is  a  necessary  Truth. 
— In  the  series  of  discussions  in  which  we  are  here 
engaged,  our  main  business  is  to  ascertain  the  nature  and 
grounds  of  the  certainty  of  scientific  truths.  We  have, 
therefore,  to  ask  whether  this  proposition,  the  parallelo 
gram  of  forces,  be  a  necessary  truth  ;  and  if  so,  on  what 
grounds  its  necessity  ultimately  rests.  We  shall  find 
that  this,  like  the  other  fundamental  doctrines  of  Statics, 
justly  claims  a  demonstrative  certainty.  Daniel  Ber 
noulli,  in  1726,  gave  the  first  proof  of  this  important 
proposition  on  pure  statical  principles ;  and  thus,  as  he 
says*,  "proved  that  statical  theorems  are  not  less 
necessarily  true  than  geometrical  are."  If  we  examine 
this  proof  of  Bernoulli,  in  order  to  discover  what  are 
the  principles  on  which  it  rests,  we  shall  find  that  the 
reasoning  employs  in  its  progress  such  axioms  as  this ; — 
That  if  from  forces  which  arc  in  equilibrium  at  a  point 

*   Comm.  Pctrop.  Vol.  i. 


be  taken  away  other  forces  which  are  in  equilibrium  at 
the  same  point,  the  remainder  will  be  in  equilibrium ; 
and  generally ; — That  if  forces  can  be  resolved  into  other 
equivalent  forces,  these  may  be  separated,  grouped,  and 
recombined,  in  any  new  manner,  and  the  result  will  still 
be  identical  with  what  it  was  at  first.  Thus  in  Ber 
noulli's  proof,  the  two  forces  to  be  compounded  are  repre 
sented  by  P  and  Q  ;  p  is  resolved  into  two  other  forces,  x 
and  u ;  and  Q,  into  two  others,  Y  and  v,  under  certain 
conditions.  It  is  then  assumed  that  these  forces  may  be 
grouped  into  the  pairs  x,  Y,  and  u,  v :  and  when  it  has 
been  shown  that  x  and  Y  are  in  equilibrium,  they  may,  by 
what  has  been  said,  be  removed,  and  the  forces,  p,  Q,  are 
equivalent  to  u,  v ;  which,  being  in  the  same  direction 
by  the  course  of  the  construction,  have  a  result  equal  to 
their  sum. 

It  is  clear  that  the  principles  here  assumed  are 
genuine  axioms,  depending  upon  our  conception  of  the 
nature  of  equivalence  of  forces,  and  upon  their  being 
capable  of  addition  and  composition.  If  the  forces  P,  Q, 
be  equivalent  to  forces  x,  u,  Y,  v,  they  are  equivalent  to 
these  forces  added  and  compounded  in  any  order ;  just 
as  a  geometrical  figure  is,  by  our  conception  of  space, 
equivalent  to  its  parts  added  together  in  any  order.  The 
apprehension  of  forces  as  having  magnitude,  as  made 
up  of  parts,  as  capable  of  composition,  leads  to  such 
axioms  in  Statics,  in  the  same  manner  as  the  like 
apprehension  of  space  leads  to  the  axioms  of  Geometry. 
And  thus  the  truths  of  Statics,  resting  upon  such  founda 
tions,  are  independent  of  experience  in  the  same  manner 
in  which  geometrical  truths  are  so. 

The  proof  of  the  parallelogram  of  forces  thus  given 
by  Daniel  Bernoulli,  as  it  was  the  first,  is  also  one  of 
the  most  simple  proofs  of  that  proposition  which  have 
been  devised  up  to  the  present  day.  Many  other  demon- 


strations,  however,  have  been  given  of  the  same  proposi 
tion.  Jacobi,  a  German  mathematician,  has  collected 
and  examined  eighteen  of  these"'.  They  all  depend 
either  upon  such  principles  as  have  just  been  stated ; 
That  forces  may  in  every  way  be  replaced  by  those  which 
are  equivalent  to  them ; — or  else  upon  those  previously 
stated,  the  doctrine  of  the  lever,  and  the  transfer  of  a 
force  from  one  point  to  another  of  its  direction.  In 
either  case,  they  are  necessary  results  of  our  statical  con 
ceptions,  independent  of  any  observed  laws  of  motion, 
and  indeed,  of  the  conception  of  actual  motion  altogether. 
There  is  another  class  of  alleged  proofs  of  the  paral 
lelogram  of  forces,  which  involve  the  consideration  of 
the  motion  produced  by  the  forces.  But  such  reasonings 
are,  in  fact,  altogether  irrelevant  to  the  subject  of  Statics. 
In  that  science,  forces  are  not  measured  by  the  motion 
which  they  produce,  but  by  the  forces  which  they  will 
balance,  as  we  have  already  seen.  The  combination  of 
two  forces  employed  in  producing  motion  in  the  same 
body,  either  simultaneously  or  successively,  belongs  to 
that  part  of  Mechanics  which  has  motion  for  its  subject, 
and  is  to  be  considered  in  treating  of  the  laws  of  motion. 
The  composition  of  motion,  (as  when  a  man  moves  in  a 
ship  while  the  ship  moves  through  the  water,)  has  con 
stantly  been  confounded  with  the  composition  of  force. 
But  though  it  has  been  done  by  very  eminent  mathe 
maticians,  it  is  quite  necessary  for  us  to  keep  the  two 
subjects  distinct,  in  order  to  see  the  real  nature  of  the 
evidence  of  truth  in  either  case.  The  conditions  of  equi 
librium  of  two  forces  on  a  lever,  or  of  three  forces  at 

*  These  arc  by  the  following  mathematicians;  D.  Bernoulli 
(1726);  Lambert  (1771);  Scarella  (175fi);  Venini  (1 704);  Araldi 
(180(>);  Wachter  (IHlii);  Kajstner  ;  Marini ;  Eytelwein  ;  Salimbeni ; 
Duchayla;  two  different  proofs  by  Foncenex  (17GO) ;  three  by 
D'Alembert;  and  those  of  Laplace  and  M.  Poisson. 


a  point,  can  be  established  without  any  reference  what 
ever  to  any  motions  which  the  forces  might,  under  other 
circumstances,  produce.  And  because  this  can  be  done, 
to  do  so  is  the  only  scientific  procedure.  To  prove  such 
propositions  by  any  other  course,  would  be  to  support 
truth  by  extraneous  and  inconclusive  reasons;  which 
would  be  foreign  to  our  purpose,  since  we  seek  not  only 
knowledge,  but  the  grounds  of  our  knowledge. 

10.    The  Center  of  gravity  seeks  the  lowest  place. — 
The  principles  which  we  have  already  mentioned  afford 
a  sufficient  basis  for  the  science  of  Statics  in  its  most 
extensive  and  varied  applications ;  and  the  conditions  of 
equilibrium  of  the  most  complex  combinations  of  ma 
chinery  may  be  deduced  from  these  principles  with  a 
rigour  not  inferior  to  that  of  geometry.     But  in  some  of 
the  more  complex  cases,  the   results  of  long  trains  of 
reasoning  may  be  foreseen,  in  virtue  of  certain  maxims 
which  appear  to  us  self-evident,  although  it  may  not  be 
easy  to  trace  the  exact  dependence  of  these  maxims  upon 
our  fundamental  conceptions  of  force  and  matter.     Of 
this  nature  is  the  maxim  now  stated ; — That  in  any  com 
bination  of  matter  any  how   supported,   the  Center  of 
Gravity  will  descend  into  the  lowest  position  which  the 
connexion  of  the  parts  allows  it  to  assume  by  descend 
ing.    It  is  easily  seem  that  this  maxim  carries  to  a  much 
greater  extent  the  principle   which   the   Greek  mathe 
maticians   assumed,   that   every  body  has  a  Center    of 
Gravity,  that  is,  a  point  in  which,  if  the  whole  matter  of 
the  body  be  collected,  the  effect  will  remain  unchanged. 
For  the  Greeks  asserted  this  of  a  single  rigid  mass  only ; 
whereas,  in  the  maxim  now  under  our  notice,  it  is  asserted 
of  any  masses,  connected  by  strings,  rods,  joints,  or  in 
any  manner.     We  have  already  seen  that  more  modern 
writers  on  mechanics,  desirous  of  assuming  as  funda 
mental  no  wider  principles  than  are  absolutely  necessary, 


have  not  adopted  the  Greek  axiom  in  all  its  generality, 
but  have  only  asserted  that  two  equal  weights  have  a 
center  of  gravity  midway  between  them.  Yet  the  prin 
ciple  that  every  body,  however  irregular,  has  a  center  of 
gravity,  and  will  be  supported  if  that  center  is  supported, 
and  not  otherwise,  is  so  far  evident,  that  it  might  be 
employed  as  a  fundamental  truth,  if  we  could  not  resolve 
it  into  any  simpler  truths :  and,  historically  speaking,  it 
was  assumed  as  evident  by  the  Greeks.  In  like  manner 
the  still  wider  principle,  that  a  collection  of  bodies,  as, 
for  instance,  a  flexible  chain  hanging  upon  one  or  more 
supports,  has  a  center  of  gravity ;  and  that  this  point 
will  descend  to  the  lowest  possible  situation,  as  a  single 
body  would  do,  has  been  adopted  at  various  periods  in 
the  history  of  mechanics ;  and  especially  at  conjunctures 
when  mathematical  philosophers  have  had  new  and  dif 
ficult  problems  to  contend  with.  For  in  almost  every 
instance  it  has  only  been  by  repeated  struggles  that  phi 
losophers  have  reduced  the  solution  of  such  problems  to 
a  clear  dependence  upon  the  most  simple  axioms. 

11.  Stevinuss  Proof  for  Oblique  Forces. — We  have 
an  example  of  this  mode  of  dealing  with  problems,  in 
Stevinus's  mode  of  reasoning  concerning  the  Inclined 
Plane ;  which,  as  we  have  stated  in  the  History  of  Me 
chanics,  was  the  first  correct  published  solution  of  that 
problem.  Stevinus  supposes  a  loop  of  chain,  or  a  loop 
of  string  loaded  with  a  series  of  equal  balls  at  equal  dis 
tances,  to  hang  over  the  Inclined  Plane ;  and  his  reason 
ing  proceeds  upon  this  assumption, — That  such  a  loop 
so  hanging  will  find  a  certain  position  in  which  it  will 
rest :  for  otherwise,  says  he*,  its  motion  must  go  on  for 
ever,  which  is  absurd.  It  may  be  asked  how  this  absurd 
ity  of  a  perpetual  motion  appears ;  and  it  will  perhaps 
be  added,  that  although  the  impossibility  of  a  machine 

*  Stcvin.  Statiqnc,  Livrc  I.,  prop.  19. 


with  such  a  condition  may  be  proved  as  a  remote  result 
of  mechanical  principles,  this  impossibility  can  hardly 
be  itself  recognized  as  a  self-evident  truth.  But  to  this 
we  may  reply,  that  the  impossibility  is  really  evident  in 
the  case  contemplated  by  Stevinus ;  for  we  cannot  con 
ceive  a  loop  of  chain  to  go  on  through  all  eternity,  slid 
ing  round  and  round  upon  its  support,  by  the  effect  of 
its  own  weight.  And  the  ground  of  our  conviction  that 
this  cannot  be,  seems  to  be  this  consideration;  that  when 
the  chain  moves  by  the  effect  of  its  weight,  we  consider 
its  motion  as  the  result  of  an  effort  to  reach  some  certain 
position,  in  which  it  can  rest ;  just  as  a  single  ball  in 
a  bowl  moves  till  it  comes  to  rest  at  the  lowest  point 
of  the  bowl.  Such  an  effect  of  weight  in  the  chain,  we 
may  represent  to  ourselves  by  conceiving  all  the  matter 
of  the  chain  to  be  collected  in  one  single  point,  and  this 
single  heavy  point  to  hang  from  the  support  in  some  way 
or  other,  so  as  fitly  to  represent  the  mode  of  support  of 
the  chain.  In  whatever  manner  this  heavy  point  (the 
center  of  gravity  of  the  chain)  be  supported  and  con 
trolled  in  its  movements,  there  will  still  be  some  position 
of  rest  which  it  will  seek  and  find.  And  thus  there  will 
be  some  corresponding  position  of  rest  for  the  chain  ;  and 
the  interminable  shifting  from  one  position  to  another, 
with  no  disposition  to  rest  in  any  position,  cannot  exist. 

Thus  the  demonstration  of  the  property  of  the 
Inclined  Plane  by  Stevinus,  depends  upon  a  principle 
which,  though  far  from  being  the  simplest  of  those  to 
which  the  case  can  be  reduced,  is  still  both  true  and 
evident :  and  the  evidence  of  this  principle,  depending 
upon  the  assumption  of  a  center  of  gravity,  is  of  the 
same  nature  as  the  evidence  of  the  Greek  statical  demon 
strations,  the  earliest  real  advances  in  the  science. 

12.  Principle  of  Virtual  Velocities. — We  have 
referred  above  to  an  assertion  often  made,  that  we 


may,  from  the  simple  principles  of  Mechanics,  demon 
strate  the  impossibility  of  a  perpetual  motion.  In  reality, 
however,  the  simplest  proof  of  that  impossibility,  in 
a  machine  acted  upon  by  weight  only,  arises  from  the 
very  maxim  above  stated,  that  the  center  of  gravity  seeks 
and  finds  the  lowest  place ;  or  from  some  similar  propo 
sition.  For  if,  as  is  done  by  many  writers,  we  profess 
to  prove  the  impossibility  of  a  perpetual  motion  by  means 
of  that  proposition  which  includes  the  conditions  of  equi 
librium,  and  is  called  the  Principle  of  Virtual  Velocities*, 
we  are  under  the  necessity  of  first  proving  in  a  general 
manner  that  principle.  And  if  this  be  done  by  a  mere 
enumeration  of  cases,  (as  by  taking  those  five  cases  which 
are  called  the  Mechanical  Powers,}  there  may  remain 
some  doubts  whether  the  enumeration  of  possible  mecha 
nical  combinations  be  complete.  Accordingly,  some  writers 
have  attempted  independent  and  general  proofs  of  the 
Principle  of  Virtual  Velocities;  and  these  proofs  rest 
upon  assumptions  of  the  same  nature  as  that  now  under 
notice.  This  is,  for  example,  the  case  with  Lagrange's 
proof,  which  depends  upon  what  he  calls  the  Principle 
of  Pulleys.  For  this  principle  is, — That  a  weight  any 
how  supported,  as  by  a  string  passing  round  any  number 
of  pulleys  any  how  placed,  will  be  at  rest  then  only, 
when  it  cannot  get  lower  by  any  small  motion  of  the 
pulleys.  And  thus  the  maxim  that  a  weight  will  descend 
if  it  can,  is  assumed  as  the  basis  of  this  proof. 

There  is,  as  we  have  said,  no  need  to  assume  such 
principles  as  these  for  the  foundation  of  our  mechanical 
science.  But  it  is,  on  various  accounts,  useful  to  direct 
our  attention  to  those  cases  in  which  truths,  apprehended 
at  first  in  a  complex  and  derivative  form,  have  after 
wards  been  reduced  to  their  simpler  elements ; — in  which, 
also,  sagacious  and  inventive  men  have  fixed  upon  those 

*  See  Hist.  I»d.  Sci.,  B.  vi.  c.  ii.  sect.  4. 
VOL.   I.     \V.  P.  I' 


truths  as  self-evident,  which  now  appear  to  us  only  cer 
tain  in  virtue  of  demonstration.  In  these  cases  we  can 
hardly  doubt  that  such  men  were  led  to  assert  the 
doctrines  which  they  discovered,  not  by  any  capricious 
conjecture  or  arbitrary  selection,  but  by  having  a  keener 
and  deeper  insight  than  other  persons  into  the  relations 
which  were  the  object  of  their  contemplation ;  and  in  the 
science  now  spoken  of,  they  were  led  to  their  assump 
tions  by  possessing  clearly  and  distinctly  the  conceptions 
of  mechanical  cause  and  effect, — action  and  reaction, — 
force,  and  the  nature  of  its  operation. 

13.  Fluids  2^'ess  Equally  in  all  Directions. — The 
doctrines  which  concern  the  equilibrium  of  fluids  depend 
on  principles  no  less  certain  and  simple  than  those  which 
refer  to  the  equilibrium  of  solid  bodies ;  and  the  Greeks, 
who,  as  we  have  seen,  obtained  a  clear  view  of  some  of 
the  principles  of  Statics,  also  made  a  beginning  in  the 
kindred  subject  of  Hydrostatics.  We  still  possess  a  trea 
tise  of  Archimedes  On  Floating  Bodies,  which  contains 
correct  solutions  of  several  problems  belonging  to  this 
subject,  and  of  some  which  are  by  no  means  easy.  In 
this  treatise,  the  fundamental  assumption  is  of  this  kind  : 
"Let  it  be  assumed  that  the  nature  of  a  fluid  is  such, 
that  the  parts  which  are  less  pressed  yield  to  those  which 
are  more  pressed."  In  this  assumption  or  axiom  it  is 
implied  that  a  pressure  exerted  upon  a  fluid  in  one  direc 
tion  produces  a  pressure  in  another  direction ;  thus,  the 
weight  of  the  fluid  which  arises  from  a  downward  force 
produces  a  lateral  pressure  against  the  sides  of  the  con 
taining  vessel.  Not  only  does  the  pressure  thus  diverge 
from  its  original  direction  into  all  other  directions,  but  the 
pressure,  is  in  all  directions  exactly  equal,  an  equal  extent 
of  the  fluid  being  taken.  This  principle,  which  was  in 
volved  in  the  reasoning  of  Archimedes,  is  still  to  the 
present  day  the  basis  of  all  hydrostatical  treatises,  and  is 


expressed,  as  above,  by  saying  that  fluids  press  equally 
in  all  directions. 

Concerning  this,  as  concerning  previously-noticed 
principles,  we  have  to  ask  whether  it  can  rightly  be  said 
to  be  derived  from  experience.  And  to  this  the  answer 
must  still  be,  as  in  the  former  cases,  that  the  proposition 
is  not  one  borrowed  from  experience  in  any  usual  or 
exact  sense  of  the  phrase.  I  will  endeavour  to  illustrate 
this.  There  are  many  elementary  propositions  in  phy 
sics,  our  knowledge  of  which  indisputably  depends  upon 
experience ;  and  in  these  cases  there  is  no  difficulty  in 
seeing  the  evidence  of  this  dependence.  In  such  cases, 
the  experiments  which  prove  the  law  are  prominently 
stated  in  treatises  upon  the  subject :  they  are  given  with 
exact  measures,  and  with  an  account  of  the  means  by 
which  errors  were  avoided :  the  experiments  of  more 
recent  times  have  either  rendered  more  certain  the  law 
originally  asserted,  or  have  pointed  out  some  correction 
of  it  as  requisite  :  and  the  names,  both  of  the  discoverers 
of  the  law  and  of  its  subsequent  reformers,  are  well 
known.  For  instance,  the  proposition  that  "  The  elastic 
force  of  air  varies  as  the  density,"  was  first  proved  by 
Boyle,  by  means  of  operations  of  which  the  detail  is  given 
in  his  Defence  of  his  Pneumatical  Experiments* ;  and 
by  Marriotte  in  his  Traite  de  VEquilibre  des  Liquides, 
from  whom  it  has  generally  been  termed  Marriotte's  law. 
After  being  confirmed  by  many  other  experimenters, 
this  law  was  suspected  to  be  slightly  inaccurate,  and  a 
commission  of  the  French  Academy  of  Sciences  was 
;i|>j)ointed,  consisting  of  several  distinguished  philoso- 
phersf,  to  ascertain  the  truth  or  falsehood  of  this  suspicion. 

*  Shaw's  Boyle,  Vol.  ii.  p.  671- 

t  The  members  were  Prony,  Arago,  Ampere,  Girard,  and  Dulong. 
The  experiments  were  extended  to  a  pressure  of  twenty-seven  atmo- 
•pherea  ;  and  in  no  instance  did  the  difference  between  the  observed 

P  2 


The  result  of  their  investigations  appeared  to  be,  that 
the  law  is  exact,  as  nearly  as  the  inevitable  inaccuracies 
of  machinery  and  measures  will  allow  us  to  judge.  Here 
we  have  an  example  of  a  law  which  is  of  the  simplest 
kind  and  form ;  and  which  yet  is  not  allowed  to  rest 
upon  its  simplicity  or  apparent  probability,  but  is  rigor 
ously  tested  by  experience.  In  this  case,  the  assertion, 
that  the  law  depends  upon  experience,  contains  a  refer 
ence  to  plain  and  notorious  passages  in  the  history  of 

Now  with  regard  to  the  principle  that  fluids  press 
equally  in  all  directions,  the  case  is  altogether  different. 
It  is,  indeed,  often  asserted  in  works  on  hydrostatics, 
that  the  principle  is  collected  from  experience,  and  some 
times  a  few  experiments  are  described  as  exhibiting  its 
effect ;  but  these  are  such  as  to  illustrate  and  explain, 
rather  than  to  prove,  the  truth  of  the  principle :  they 
are  never  related  to  have  been  made  with  that  exact 
ness  of  precaution  and  measurement,  or  that  frequency 
of  repetition,  which  are  necessary  to  establish  a  purely 
experimental  truth.  Nor  did  such  experiments  occur  as 
important  steps  in  the  history  of  science.  It  does  not 
appear  that  Archimedes  thought  experiment  necessary 
to  confirm  the  truth  of  the  law  as  he  employed  it :  on 
the  contrary,  he  states  it  in  exactly  the  same  shape  as 
the  axioms  which  he  employs  in  statics,  and  even  in  geo 
metry  ;  namely,  as  an  assumption.  Nor  does  any  intel 
ligent  student  of  the  subject  find  any  difficulty  in  assent 
ing  to  this  fundamental  principle  of  hydrostatics  as  soon 
as  it  is  propounded  to  him.  Experiment  was  not  requi 
site  for  its  discovery;  experiment  is  not  necessary  for 
its  proof  at  present ;  and  we  may  add,  that  experiment, 

and  calculated  elasticity  amount  to  one-hundredth  of  the  whole ;  nor 
did  the  difference  appear  to  increase  with  the  increase  of  pressure. — 
Fechner,  Repertorium,  i.  110. 


though  it  may  make  the  proposition  more  readily  intelli 
gible,  can  add  nothing  to  our  conviction  of  its  truth 
when  it  is  once  understood. 

14.  Foundation  of  the  abore  Axiom. — But  it  will 
naturally  be  asked,  What  then  is  the  ground  of  our 
conviction  of  this  doctrine  of  the  equal  pressure  of  a 
fluid  in  all  directions'?  And  to  this  I  reply,  that  the 
reasons  of  this  conviction  are  involved  in  our  idea  of  a 
fluid,  which  is  considered  as  matter,  and  therefore  as 
capable  of  receiving,  resisting,  and  transmitting  force 
according  to  the  general  conception  of  matter;  and  which 
is  also  considered  as  matter  which  has  its  parts  perfectly 
movcable  among  one  another.  For  it  follows  from 
these  suppositions,  that  if  the  fluid  be  confined,  a  pres 
sure  which  thrusts  in  one  side  of  the  containing  vessel, 
may  cause  any  other  side  to  bulge  outwards,  if  there  be 
a  part  of  the  surface  which  has  not  strength  to  resist 
this  pressure  from  within.  And  that  this  pressure,  when 
thus  transferred  into  a  direction  different  from  the  ori 
ginal  one,  is  not  altered  in  intensity,  depends  upon  this 
consideration ;  that  any  difference  in  the  two  pressures 
would  be  considered  as  a  defect  of  jwrfect  fluidity,  since 
the  fluidity  would  be  still  more  complete,  if  this  entire 
and  undiminished  transmission  of  pressure  in  all  direc 
tions  were  supposed.  If,  for  instance,  the  lateral  pres 
sure  were  less  than  the  vertical,  this  could  be  conceived 
no  other  way  than  as  indicating  some  rigidity  or  adhesion 
of  the  parts  of  the  fluid.  When  the  fluidity  is  perfect, 
the  two  pressures  which  act  in  the  two  different  parts  of 
the  fluid  exactly  balance  each  other :  they  are  the  action 
and  the  reaction;  and  must  hence  be  equal  by  the  same 
iH-cessity  as  two  directly  opposite  forces  in  statics. 

But  it  may  be  urged,  that  even  if  we  grant  that  this 
<•<  inception  of  a  perfect  fluid,  as  a  body  which  has  its 
parts  perfectly  moveable  among  each  other,  leads  us 


necessarily  to  the  principle  of  the  equality  of  hydrostatic 
pressure  in  all  directions,  still  this  conception  itself  is 
obtained  from  experience,  or  suggested  by  observation. 
And  to  this  we  may  reply,  that  the  conception  of  a  fluid, 
as  contemplated  in  mechanical  theory,  cannot  be  said  to 
be  derived  from  experience,  except  in  the  same  manner 
as  the  conception  of  a  solid  and  rigid  body  may  be  said 
to  be  acquired  by  experience.  For  if  we  imagine  a 
vessel  full  of  small,  smooth  spherical  balls,  such  a  collec 
tion  of  balls  would  approach  to  the  nature  of  a  fluid,  in 
having  its  parts  moveable  among  each  other ;  and  would 
approach  to  perfect  fluidity,  as  the  balls  became 
smoother  and  smaller.  And  such  a  collection  of  balls 
would  also  possess  the  statical  properties  of  a  fluid ;  for 
it  would  transmit  pressure  out  of  a  vertical  into  a  lateral 
(or  any  other)  direction,  in  the  same  manner  as  a  fluid 
would  do.  And  thus  a  collection  of  solid  bodies  has 
the  same  property  which  a  fluid  has ;  and  the  science 
of  Hydrostatics  borrows  from  experience  no  principles 
beyond  those  which  are  involved  in  the  science  of 
Statics  respecting  solids.  And  since  in  this  latter  por 
tion  of  science,  as  we  have  already  seen,  none  of  the 
principles  depend  for  their  evidence  upon  any  special 
experience,  the  doctrines  of  Hydrostatics  also  are  not 
proved  by  experience,  but  have  a  necessary  truth  bor 
rowed  from  the  relations  of  our  ideas. 

It  is  hardly  to  be  expected  that  the  above  reasoning 
will,  at  first  sight,  produce  conviction  in  the  mind  of  the 
reader,  except  he  have,  to  a  certain  extent,  acquainted 
himself  with  the  elementary  doctrines  of  the  science  of 
Hydrostatics  as  usually  delivered ;  and  have  followed, 
with  clear  and  steady  apprehension,  some  of  the  trains 
of  reasoning  by  which  the  pressures  of  fluids  are  deter 
mined;  as,  for  instance,  the  explanation  of  what  is  called 
the  Hydrostatic  Parndo.r,  The  necessity  of  such  a  dis- 


cipline  in  order  that  the  reader  may  enter  fully  into  this 
part  of  our  speculations,  naturally  renders  them  less 
popular ;  but  this  disadvantage  is  inevitable  in  our  plan. 
We  cannot  expect  to  throw  light  upon  philosophy  by 
means  of  the  advances  which  have  been  made  in  the 
mathematical  and  physical  sciences,  except  we  really 
understand  the  doctrines  which  have  been  firmly  esta 
blished  in  those  sciences.  This  preparation  for  philoso 
phizing  may  be  somewhat  laborious ;  but  such  labour  is 
necessary  if  we  would  pursue  speculative  truth  with  all 
the  advantages  which  the  present  condition  of  human 
knowledge  places  within  our  reach. 

We  may  add,  that  the  consequences  to  which  we  arc 
directed  by  the  preceding  opinions,  are  of  very  great  im 
portance  in  their  bearing  upon  our  general  views  respect 
ing  human  knowledge.  I  trust  to  be  able  to  show,  that 
some  important  distinctions  arc  illustrated,  some  per 
plexing  paradoxes  solved,  and  some  large  anticipations 
of  the  future  extension  of  our  knowledge  suggested,  by 
means  of  the  conclusions  to  which  the  preceding  discus 
sions  have  conducted  us.  But  before  I  proceed  to  these 
general  topics,  I  must  consider  the  foundations  of  some 
of  the  remaining  portions  of  Mechanics. 



1.  IN  the  History  of  Mechanics,  I  have  traced  the 
steps  by  which  the  three  Laws  of  Motion  and  the  other 
principles  of  mechanics  were  discovered,  established,  and 
extended  to  the  widest  generality  of  form  and  applica 
tion.  We  have,  in  these  laws,  examples  of  principles 
which  were,  historically  speaking,  obtained  by  reference 


to  experience.  Bearing  in  mind  the  object  and  the  re 
sult  of  the  preceding  discussions,  we  cannot  but  turn 
with  much  interest  to  examine  these  portions  of  science  ; 
to  inquire  whether  there  be  any  real  difference  in  the 
grounds  and  nature  between  the  knowledge  thus  ob 
tained,  and  those  truths  which  we  have  already  contem 
plated  ;  and  which,  as  we  have  seen,  contain  their  own 
evidence,  and  do  not  require  proof  from  experiment. 

2.  The  First  Lam  of  Motion. — The  first  law  of  mo 
tion  is,  that  When  a  body  moves  not  acted  upon  by  any 
force,  it  mill  go  on  perpetually  in  a  straight  line.,  and 
with  a  uniform  Telocity.  Now  what  is  the  real  ground 
of  our  assent  to  this  proposition  ?  That  it  is  not  at  first 
sight  a  self-evident  truth,  appears  to  be  clear ;  since  from 
the  time  of  Aristotle  to  that  of  Galileo  the  opposite 
assertion  was  held  to  be  true ;  and  it  was  believed  that 
all  bodies  in  motion  had,  by  their  own  nature,  a  constant 
tendency  to  move  more  and  more  slowly,  so  as  to  stop  at 
last.  This  belief,  indeed,  is  probably  even  now  enter 
tained  by  most  persons,  till  their  attention  is  fixed  upon 
the  arguments  by  which  the  first  law  of  motion  is  esta 
blished.  It  is,  however,  not  difficult  to  lead  any  person 
of  a  speculative  habit  of  thought  to  see  that  the  retard 
ation  which  constantly  takes  place  in  the  motion  of  all 
bodies  when  left  to  themselves,  is,  in  reality,  the  effect 
of  extraneous  forces  which  destroy  the  velocity.  A  top 
ceases  to  spin  because  the  friction  against  the  ground 
and  the  resistance  of  the  air  gradually  diminish  its  mo 
tion,  and  not  because  its  motion  has  any  internal  prin 
ciple  of  decay  or  fatigue.  This  may  be  shown,  and  was, 
in  fact,  shown  by  Hooke  before  the  Royal  Society,  at  the 
time  when  the  laws  of  motion  were  still  under  discus 
sion,  by  means  of  experiments  in  which  the  weight  of 
the  top  is  increased,  and  the  resistance  to  motion  offered 
by  its  support,  is  diminished ;  for  by  such  contrivances 

>  .• 

its  motion  is  made  to  continue  much  longer  than  it 
would  otherwise  do.  And  by  experiments  of  this  nature, 
although  we  can  never  remove  the  whole  of  the  external 
impediments  to  continued  motion,  and  although,  conse 
quently,  there  will  always  be  some  retardation  ;  and  an 
end  of  the  motion  of  a  body  left  to  itself,  however  long 
it  may  be  delayed,  must  at  last  come ;  yet  we  can  esta 
blish  a  conviction  that  if  all  resistance  could  de  removed, 
there  would  be  no  diminution  of  velocity,  and  thus  the 
motion  would  go  on  for  ever. 

If  we  call  to  mind  the  axioms  which  we  formerly 
stated,  as  containing  the  most  important  conditions 
involved  in  the  idea  of  Cause,  it  will  be  seen  that  our 
conviction  in  this  case  depends  upon  the  first  axiom  of 
Causation,  that  nothing  can  happen  without  a  cause. 
Every  change  in  the  velocity  of  the  moving  body  must 
have  a  cause ;  and  if  the  change  can,  in  any  manner,  be 
referred  to  the  presence  of  other  bodies,  these  are  said 
to  exert  force  upon  the  moving  body:  and  the  conception 
of  force  is  thus  evolved  from  the  general  idea  of  cause. 
Force  is  any  cause  which  has  motion,  or  change  of 
motion,  for  its  effect ;  and  thus,  all  the  change  of  velocity 
of  a  body  which  can  be  referred  to  extraneous  bodies, — as 
the  air  which  surrounds  it,  or  the  support  on  which  it 
rests, — is  considered  as  the  effect  of  forces;  and  this 
consideration  is  looked  upon  as  explaining  the  difference 
between  the  motion  which  really  takes  places  in  the  expe 
riment,  and  that  motion  which,  as  the  law  asserts,  would 
take  place  if  the  body  were  not  acted  on  by  any  forces. 

Thus  the  truth  of  the  first  law  of  motion  depends 
upon  the  axiom  that  no  change  can  take  place  without  a 
cause;  and  follows  from  the  definition  of  force,  if  we  sup 
pose  that  there  can  be  none  but  an  external  cause  of 
change.  But  in  order  to  establish  the  law,  it  was  neces 
sary  further  to  be  assured  that  tlr  re  is  no  internal  cause 


of  change  of  velocity  belonging  to  all  matter  whatever, 
and  operating  in  such  a  manner  that  the  mere  progress 
of  time  is  sufficient  to  produce  a  diminution  of  velocity 
in  all  moving  bodies.  It  appears  from  the  history  of 
mechanical  science,  that  this  latter  step  required  a  refer 
ence  to  observation  and  experiment ;  and  that  the  first 
law  of  motion  is  so  far,  historically  at  least,  dependent 
upon  our  experience. 

But  notwithstanding  this  historical  evidence  of  the 
need  which  we  have  of  a  reference  to  observed  facts,  in 
order  to  place  this  first  law  of  motion  out  of  doubt,  it  has 
been  maintained  by  very  eminent  mathematicians  and 
philosophers,  that  the  law  is,  in  truth,  evident  of  itself, 
and  does  not  really  rest  upon  experimental  proof.  Such, 
for  example,  is  the  opinion  of  D'Alembert*,  who  offers 
what  is  called  an  d  priori  proof  of  this  law ;  that  is,  a 
demonstration  derived  from  our  ideas  alone.  When  a 
body  is  put  in  motion,  either,  he  says,  the  cause  which 
puts  it  in  motion  at  first,  suffices  to  make  it  move  one 
foot,  or  the  continued  action  of  the  cause  during  this  foot 
is  requisite  for  the  motion.  In  the  first  case,  the  same 
reason  which  made  the  body  proceed  to  the  end  of  the 
first  foot  will  hold  for  its  going  on  through  a  second, 
a  third,  a  fourth  foot,  and  so  on  for  any  number.  In 
the  second  case,  the  same  reason  which  made  the  force 
continue  to  act  during  the  first  foot,  will  hold  for  its 
acting,  and  therefore  for  the  body  moving  during  each 
succeeding  foot.  And  thus  the  body,  once  beginning  to 
move,  must  go  on  moving  for  ever. 

It  is  obvious  that  we  might  reply  to  this  argument, 
that  the  reasons  for  the  body  proceeding  during  each 
succeeding  foot  may  not  necessarily  be  all  the  same ;  for 
among  these  reasons  may  be  the  time  which  has  elapsed; 
and  thus  the  velocity  may  undergo  a  change  as  the  time 


proceeds :  and  we  require  observation  to  inform  us  that 
it  does  not  do  so. 

Professor  Playfair  has  presented  nearly  the  same 
argument,  although  in  a  different  and  more  mathematical 
form*.  If  the  velocity  change,  says  he,  it  must  change 
according  to  some  expression  of  calculation  depending 
upon  the  time,  or,  in  mathematical  language,  must  be  a 
function  of  the  time.  If  the  velocity  diminish  as  the 
time  increases,  this  may  be  expressed  by  stating  the  velo 
city  in  each  case  as  a  certain  number,  from  which  another 
quantity,  or  term,  increasing  as  the  time  increases,  is 
subtracted.  But,  Playfair  adds,  there  is  no  condition 
involved  in  the  nature  of  the  case,  by  which  the  coeffi 
cients,  or  numbers  which  are  to  be  employed,  along  with 
the  number  representing  the  time,  in  calculating  this 
second  term,  can  be  determined  to  be  of  one  magnitude 
rather  than  of  any  other.  Therefore  he  infers  there  can 
be  no  such  coefficients,  and  that  the  velocity  is  in  each 
case  equal  to  some  constant  number,  independent  of  the 
time ;  and  is  therefore  the  same  for  all  times. 

In  reply  to  this  we  may  observe,  that  the  circum 
stance  of  our  not  seeing  in  the  nature  of  the  case  any 
thing  which  determines  for  us  the  coefficients  above- 
spoken  off,  cannot  prove  that  they  have  not  some  certain 
value  in  nature.  We  do  not  see  in  the  nature  of  the 
case  anything  which  should  determine  a  body  to  fall  six 
teen  feet  in  a  second  of  time,  rather  than  one  foot  or  one 
hundred  feet :  yet  in  fact  the  space  thus  run  through  by 
falling  bodies  is  determined  to  a  certain  magnitude.  It 
would  be  easy  to  assign  a  mathematical  expression  for 
the  velocity  of  a  body,  implying  that  one-hundredth  of  the 
velocity,  or  any  other  fraction,  is  lost  in  each  second  f: 

*  Outlines,  &c.,  j>.  2(5. 

+  This  would  IK;  the  case,  if,  /  being  the  number  of  seconds  elapsed, 


and  where  is  the  absurdity  of  supposing  such  an  expres 
sion  really  to  represent  the  velocity  ? 

Most  modern  writers  on  mechanics  have  embraced 
the  opposite  opinion,  and  have  ascribed  our  knowledge 
of  this  first  law  of  motion  to  experience.  Thus  M. 
Poisson,  one  of  the  most  eminent  of  the  mathematicians 
who  have  written  on  this  subject,  says*,  "  We  cannot 
affirm  a  priori  that  the  velocity  communicated  to  a  body 
will  not  become  slower  and  slower  of  itself,  and  end  by 
being  entirely  extinguished.  It  is  only  by  experience 
and  induction  that  this  question  can  be  decided." 

Yet  it  cannot  be  denied  that  there  is  much  force  in 
those  arguments  by  which  it  is  attempted  to  shew  that 
the  First  Law  of  Motion,  such  as  we  find  it,  is  more 
consonant  to  our  conceptions  than  any  other  would  be. 
The  Law,  as  it  exists,  is  the  most  simple  that  we  can 
conceive.  Instead  of  having  to  determine  by  experi 
ments  what  is  the  law  of  the  natural  change  of  velocity, 
we  find  the  Law  to  be  that  it  does  not  change  at  all.  To  a 
certain  extent,  the  Law  depends  upon  the  evident  axiom, 
that  no  change  can  take  place  without  a  cause.  But 
the  question  further  occurs,  whether  the  mere  lapse  of 
time  may  not  be  a  cause  of  change  of  velocity.  In  order 
to  ensure  this,  we  have  recourse  to  experiment  ;  and  the 
result  is  that  time  alone  does  not  produce  any  such 
change.  In  addition  to  the  conditions  of  change  which 
we  collect  from  our  own  Ideas,  we  ask  of  Experience  what 
other  conditions  and  circumstances  she  has  to  offer  ;  and 
the  answer  is,  that  she  can  point  out  none.  When  we 
have  removed  the  alterations  which  external  causes,  in 

and  C  some  constant  quantity,  the  velocity  were  expressed  by  this 
mathematical  formula, 

Poisson,  Dynamique,   Ed.  2,  Art.  113. 


our  very  conception  of  them,  occasion,  there  are  no 
longer  any  alterations.  Instead  of  having  to  guide  our 
selves  by  experience,  we  learn  that  on  this  subject  she 
has  nothing  to  tell  us.  Instead  of  having  to  take  into 
account  a  number  of  circumstances,  we  find  that  we  have 
only  to  reject  all  circumstances.  The  velocity  of  a  body 
remains  unaltered  by  time  alone,  of  whatever  kind  the 
body  itself  be. 

But  the  doctrine  that  time  alone  is  not  a  cause  of 
change  of  velocity  in  any  body  is  further  recommended 
to  us  by  this  consideration ; — that  time  is  conceived  by 
us  not  as  a  cause,  but  only  as  a  condition  of  other  causes 
producing  their  effects.  Causes  operate  in  time ;  but  it 
is  only  when  the  cause  exists,  that  the  lapse  of  time  can 
give  rise  to  alterations.  When  therefore  all  external 
causes  of  change  of  velocity  are  supposed  to  be  removed, 
the  velocity  must  continue  identical  with  itself,  whatever 
the  time  which  elapses.  An  eternity  of  negation  can 
produce  no  positive  result. 

Thus,  though  the  discovery  of  the  First  Law  of 
Motion  was  made,  historically  speaking,  by  means  of 
experiment,  we  have  now  attained  a  point  of  view  in 
which  we  see  that  it  might  have  been  certainly  known 
to  be  true  independently  of  experience.  This  law  in  its 
ultimate  form,  when  completely  simplified  and  steadily 
contemplated,  assumes  the  character  of  a  self-evident 
truth.  We  shall  find  the  same  process  to  take  place  in 
other  instances.  And  this  feature  in  the  progress  of 
science  will  hereafter  be  found  to  suggest  very  important 
views  with  regard  both  to  the  nature  and  prospects  of 
our  knowledge. 

3.  Gravity  is  a  Uniform  Force. — We  shall  find 
observations  of  the  same  kind  offering  themselves  in  a 
manner  more  or  less  obvious,  with  regard  to  the  other 
principles  of  Dynamics.  The  determination  of  the  laws 


according  to  which  bodies  fall  downwards  by  the  com 
mon  action  of  gravity,  has  already  been  noticed  in  the 
History  of  Mechanics*,  as  one  of  the  earliest  positive 
advances  in  the  doctrine  of  motion.  These  laws  were 
first  rightly  stated  by  Galileo,  and  established  by  rea 
soning  and  by  experiment,  not  without  dissent  and  con 
troversy.  The  amount  of  these  doctrines  is  this :  That 
gravity  is  a  uniform  accelerating  force ;  such  a  uniform 
force  having  this  for  its  character,  that  it  makes  the 
velocity  increase  in  exact  proportion  to  the  time  of 
motion.  The  relation  which  the  spaces  described  by  the 
body  bear  to  the  times  in  which  they  are  described,  is 
obtained  by  mathematical  deduction  from  this  definition 
of  the  force. 

The  clear  Definition  of  a  uniform  accelerating  force, 
and  the  Proposition  that  gravity  is  such  a  force,  were 
co-ordinate  and  contemporary  steps  in  this  discovery. 
In  defining  accelerating  force,  reference,  tacit  or  ex 
press,  was  necessarily  made  to  the  second  of  the  general 
axioms  respecting  causation, — That  causes  are  measured 
by  their  effects.  Force,  in  the  cases  now  under  our 
notice,  is  conceived  to  be,  as  we  have  already  stated, 
(p.  217,)  any  cause  which,  acting  from  without,  changes 
the  motion  of  a  body.  It  must,  therefore,  in  this  accep 
tation,  be  measured  by  the  magnitude  of  the  changes 
which  are  produced.  But  in  what  manner  the  changes 
of  motion  are  to  be  employed  as  the  measures  of  force,  is 
learnt  from  observation  of  the  facts  which  we  see  taking 
place  in  the  world.  Experience  interprets  the  axiom  of 
causation,  from  which  otherwise  we  could  not  deduce 
any  real  knowledge.  We  may  assume,  in  virtue  of  our 
general  conceptions  of  force,  that  under  the  same  cir 
cumstances,  a  greater  change  of  motion  implies  a  greater 
force  producing  it ;  but  what  are  we  to  expect  when  the 

*  Hist.  hid.  Sci.,  B   vi.  c.  ii.  sect.  2. 


circumstances  change  ?  The  weight  of  a  body  makes  it 
fall  from  rest  at  first,  and  causes  it  to  move  more  quickly 
as  it  descends  lower.  We  may  express  this  by  saying, 
that  gravity,  the  universal  force  which  makes  all  terres 
trial  bodies  fall  when  not  supported,  by  its  continuous 
action  first  gives  velocity  to  the  body  when  it  has  none, 
and  afterwards  adds  velocity  to  that  which  the  body 
already  has.  But  how  is  the  velocity  added  propor 
tioned  to  the  velocity  which  already  exists?  Force 
acting  on  a  body  at  rest,  and  on  a  body  in  motion, 
appears  under  very  different  conditions ; — how  are  the 
effects  related  ?  Let  the  force  be  conceived  to  be  in  both 
cases  the  same,  since  force  is  conceived  to  depend  upon 
the  extraneous  bodies,  and  not  upon  the  condition  of  the 
moving  mass  itself.  But  the  force  being  the  same,  the 
effects  may  still  be  different.  It  is  at  first  sight  con 
ceivable  that  the  body,  acted  upon  by  the  same  gravity, 
may  receive  a  less  addition  of  velocity  when  it  is  already 
moving  in  the  direction  in  which  this  gravity  impels  it  ; 
for  if  we  ourselves  push  a  body  forwards,  we  can  produce 
little  additional  effect  upon  it  when  it  is  already  moving 
rapidly  away  from  us.  May  it  not  be  true,  in  like  man 
ner,  that  although  gravity  be  always  the  same  force,  its 
effect  depends  upon  the  velocity  which  the  body  under 
its  influence  already  possesses  ? 

Observation  and  reasoning  combined,  as  we  have 
said,  enabled  Galileo  to  answer  these  questions.  He  as 
serted  and  proved  that  we  may  consistently  and  properly 
measure  a  force  by  the  velocity  which  is  by  it  generated 
in  a  body,  in  some  certain  time,  as  one  second ;  and 
further,  that  if  we  adopt  this  measure,  gravity  will  be  a 
force  of  the  same  value  under  all  circumstances  of  the 
body  which  it  affects;  since  it  appeared  that,  in  fact,  a 
falling  body  does  receive  equal  increments  of  velocity 
in  equal  times  from  first  to  last. 


If  it  be  asked  whether  we  could  have  known,  anterior 
to,  or  independent  of,  experiment,  that  gravity  is  a  uni 
form  force  in  the  sense  thus  imposed  upon  the  term ; 
it  appears  clear  that  we  must  reply,  that  we  could  not 
have  attained  to  such  knowledge,  since  other  laws  of  the 
motion  of  bodies  downwards  are  easily  conceivable,  and 
nothing  but  observation  could  inform  us  that  one  of 
these  laws  does  not  prevail  in  fact.  Indeed,  we  may  add, 
that  the  assertion  that  the  force  of  gravity  is  uniform,  is 
so  far  from  being  self-evident,  that  it  is  not  even  true ; 
for  gravity  varies  according  to  the  distance  from  the 
center  of  the  earth ;  and  although  this  variation  is  so 
small  as  to  be,  in  the  case  of  falling  bodies,  imperceptible, 
it  negatives  the  rigorous  uniformity  of  the  force  as  com 
pletely,  though  not  to  the  same  extent,  as  if  the  weight 
of  a  body  diminished  in  a  marked  degree,  when  it  was 
carried  from  the  lower  to  the  upper  room  of  a  house.  It 
cannot,  then,  be  a  truth  independent  of  experience,  that 
gravity  is  uniform. 

Yet,  in  fact,  the  assertion  that  gravity  is  uniform  was 
assented  to,  not  only  before  it  was  proved,  but  even 
before  it  was  clearly  understood.  It  was  readily  granted 
by  all,  that  bodies  which  fall  freely  are  uniformly  accele 
rated  ;  but  while  some  held  the  opinion  just  stated,  that 
uniformly  accelerated  motion  is  that  in  which  the  velocity 
increases  in  proportion  to  the  time,  others  maintained, 
that  that  is  uniformly  accelerated  motion,  in  which  the 
velocity  increases  in  proportion  to  the  space ;  so  that,  for 
example,  a  body  in  falling  vertically  through  twenty  feet 
should  acquire  twice  as  great  a  velocity  as  one  which 
falls  through  ten  feet. 

These  two  opinions  are  both  put  forward  by  the 
interlocutors  of  Galileo's  Dialogue  on  this  subject*.  And 
the  latter  supposition  is  rejected,  the  author  showing, 
*  DialogOj  IIT.  p.  95. 


not  that  it  is  inconsistent  with  experience,  but  that  it  is 
impossible  in  itself:  inasmuch  as  it  would  inevitably  lead 
to  the  conclusion,  that  the  fall  through  a  large  and  a 
small  vertical  space  would  occupy  exactly  the  same  time. 
Indeed,  Galileo  assumes  his  definition  of  uniformly 
accelerated  motion  as  one  which  is  sufficiently  recom 
mended  by  its  own  simplicity.  "  If  we  attend  carefully," 
he  says,  "we  shall  find  that  no  mode  of  increase  of  velocity 
is  more  simple  than  that  which  adds  equal  increments  in 
equal  times.  Which  we  may  easily  understand  if  we 
consider  the  close  affinity  of  time  and  motion :  for  as  the 
uniformity  of  motion  is  defined  by  the  equality  of  spaces 
described  in  equal  times,  so  we  may  conceive  the  uni 
formity  of  acceleration  to  exist  when  equal  velocities  are 
added  in  equal  times." 

Galileo's  mode  of  supporting  his  opinion,  that  bodies 
falling  by  the  action  of  gravity  are  thus  uniformly  acce 
lerated,  consists,  in  the  first  place,  in  adducing  the 
maxim  that  nature  always  employs  the  most  simple 
means*.  But  he  is  far  from  considering  this  a  decisive 
argument.  "  I,"  says  one  of  his  speakers,  "  as  it  would 
be  very  unreasonable  in  me  to  gainsay  this  or  any  other 
definition  which  any  author  may  please  to  make,  since 
they  are  all  arbitrary,  may  still,  without  offence,  doubt 
whether  such  a  definition,  conceived  and  admitted  in  the 
abstract,  fits,  agrees,  and  is  verified  in  that  kind  of 
accelerated  motion  which  bodies  have  when  they  descend 

The  experimental  proof  that  bodies,  when  they  fall 
downwards,  are  uniformly  accelerated,  is  (by  Galileo) 
derived  from  the  inclined  plane ;  and  therefore  assumes 
the  proposition,  that  if  such  uniform  acceleration  prevail 
in  vertical  motion,  it  will  also  hold  when  a  body  is  com 
pelled  to  describe  an  oblique  rectilinear  path.  This  pro- 

*  Dialogo,  m.  j>.  91. 
VOL.  I.     W.  T.  Q 


position  may  be  shown  to  be  true,  if  (assuming  by  anti 
cipation  the  Third  Law   of  Motion,  of  which  we  shall 
shortly  have  to  speak,)  we  introduce  the  conception  of 
a  uniform  statical  force  as  the  cause  of  uniform  acce 
leration.     For   the  force   on   the  inclined  plane    bears 
a  constant   proportion   to   the   vertical  force,   and  this 
proportion  is  known  from  statical  considerations.     But 
in  the  work  of  which  we  are   speaking,    Galileo   does 
not  introduce  this  abstract  conception  of  force  as    the 
foundation    of  his  doctrines.     Instead  of  this,   he  pro 
poses,  as   a   postulate    sufficiently   evident  to   be   made 
the  basis  of  his  reasonings,  That  bodies  which  descend 
down   inclined  planes    of  different  inclinations,  but  of 
the  same  vertical  height,  all  acquire  the  same  velocity"". 
But  when  this  postulate  has  been  propounded  by  one 
of  the  persons  of  the  dialogue,  another  interlocutor  says, 
"  You   discourse  very  probably ;  but  besides  this  like 
lihood,  I  wish  to  augment  the  probability  so  far,  that 
it  shall  be  almost  as  complete  as  a  necessary  demon 
stration."     He  then  proceeds  to  describe  a  very  inge 
nious  and  simple  experiment,  which  shows  that  when  a 
body  is  made  to  swing  upwards  at  the  end  of  a  string, 
it  attains  to  the  same  height,  whatever  is  the  path  it 
follows,  so  long  as  it  starts  from  the  lowest  point  with 
the  same  velocity.     And  thus  Galileo's  postulate  is  ex 
perimentally  confirmed,  so  far  as  the  force  of  gravity  can 
be  taken  as  an  example  of  the  forces  which  the  postulate 
contemplates :  and  conversely,  gravity  is  proved  to  be  a 
uniform  force,  so  far  as  it  can  be  considered  clear  that 
the  postulate  is  true  of  uniform  forces. 

When  we  have  introduced  the  conception  and  defi 
nition  of  accelerating  force,  Galileo's  postulate,  that 
bodies  descending  down  inclined  planes  of  the  same 
vertical  height,  acquire  the  same  velocity,  may,  by  a 

*  Dialogo,  in.  p.  36. 


few  steps  of  reasoning,  be  demonstrated  to  be  true  of 
uniform  forces :  and  thus  the  proof  that  gravity,  either  in 
vertical  or  oblique  motion,  is  a  uniform  force,  is  con 
firmed  by  the  experiment  above  mentioned ;  as  it  also  is, 
on  like  grounds,  by  many  other  experiments,  made  upon 
inclined  planes  and  pendulums. 

Thus  the  propriety  of  Galileo's  conception  of  a  uni 
form  force,  and  the  doctrine  that  gravity  is  a  uniform 
force,  were  confirmed  by  the  same  reasonings  and  experi 
ments.  We  may  make  here  two  remarks ;  First,  that  the 
conception,  when  established  and  rightly  stated,  appears 
so  simple  as  hardly  to  require  experimental  proof;  a 
remark  which  we  have  already  made  with  regard  to  the 
First  Law  of  Motion :  and  Second,  that  the  discovery  of 
the  real  law  of  nature  was  made  by  assuming  proposi 
tions  which,  without  further  proof,  we  should  consider  as 
very  precarious,  and  as  far  less  obvious,  as  well  as  less 
evident,  than  the  law  of  nature  in  its  simple  form. 

4.  The  Second  Law  of  Motion. — When  a  body,  instead 
of  falling  downwards  from  rest,  is  thrown  in  any  direc 
tion,  it  describes  a  curve  line,  till  its  motion  is  stopped. 
In  this,  and  in  all  other  cases  in  which  a  body  describes 
a  curved  path  in  free  space,  its  motion  is  determined  by 
the  Second  Law  of  Motion.  The  law,  in  its  general 
form,  is  as  follows : — When  a  body  is  thus  cast  forth 
and  acted  upon  by  a  force  in  a  direction  transverse  to  its 
motion,  the  result  is,  That  there  is  combined  with  the 
motion  with  which  the  body  is  thrown,  another  motion, 
exactly  the  same  as  that  which  the  same  force  would  have 
<-(>iinnunicated  to  a  body  at  rest. 

It  will  readily  be  understood  that  the  basis  of  this 
law  is  the  axiom  already  stated,  that  effects  are  measured 
by  their  causes.  In  virtue  of  this  axiom,  the  effect  of 
gravity  acting  upon  a  body  in  a  direction  transverse  to  its 
motion,  must  measure  the  accelerative  or  deflective  force 


of  gravity  under  those  circumstances.  If  this  effect  vary 
with  the  varying  velocity  and  direction  of  the  body  thus 
acted  upon,  the  deflective  force  of  gravity  also  will  vary 
with  those  circumstances.  The  more  simple  supposition 
is,  that  the  deflective  force  of  gravity  is  the  same,  whatever 
be  the  velocity  and  direction  of  the  body  which  is  sub 
jected  to  its  influence :  and  this  is  the  supposition  which 
we  find  to  be  verified  by  facts.  For  example,  a  ball  let 
fall  from  the  top  of  a  ship's  upright  mast,  when  she  is 
sailing  steadily  forward,  will  fall  at  the  foot  of  the  mast, 
just  as  if  it  were  let  fall  while  the  ship  were  at  rest ;  thus 
showing  that  the  motion  which  gravity  gives  to  the  ball 
is  compounded  with  the  horizontal  motion  which  the  ball 
shares  with  the  ship  from  the  first.  This  general  and 
simple  conception  of  motions  as  compounded  with  one 
another,  represents,  it  is  proved,  the  manner  in  which 
the  motion  produced  by  gravity  modifies  any  other  mo 
tion  which  the  body  may  previously  have  had. 

The  discussions  which  terminated  in  the  general  re 
ception  of  this  Second  Law  of  Motion  among  mechanical 
writers,  were  much  mixed  up  with  the  arguments  for  and 
against  the  Copernican  system,  which  system  represented 
the  earth  as  revolving  upon  its  axis.  For  the  obvious 
argument  against  this  system  was,  that  if  each  point  of  the 
earth's  surface  were  thus  in  motion  from  west  to  east,  a 
stone  dropt  from  the  top  of  a  tower  would  be  left  behind, 
the  tower  moving  away  from  it :  and  the  answer  was,  that 
by  this  law  of  motion,  the  stone  would  have  the  earth's 
motion  impressed  upon  it,  as  well  as  that  motion  which 
would  arise  from  its  gravity  to  the  earth ;  and  that  the 
motion  of  the  stone  relative  to  the  tower  would  thus  be 
the  same  as  if  both  earth  and  tower  were  at  rest.  Gali 
leo  further  urged,  as  a  presumption  in  favour  of  the  opi 
nion  that  the  two  motions, — the  circular  motion  arising 
from  the  rotation  of  the  earth,  and  the  downward  motion 


arising  from  the  gravity  of  the  stone,  would  be  com 
pounded  in  the  way  we  have  described,  (neither  of  them 
disturbing  or  diminishing  the  other,) — that  the  first 
motion  was  in  its  own  nature  not  liable  to  any  change  or 
diminution*,  as  we  learn  from  the  First  Law  of  Motion. 
Nor  was  the  subject  lightly  dismissed.  The  experiment 
of  the  stone  let  fall  from  the  top  of  the  mast  was  made 
in  various  forms  by^'Gassendi ;  and  in  his  Epistle,  De 
Motu  impresso  a  Motore  translate,  the  rule  now  in  ques 
tion  is  supported  by  reference  to  these  experiments.  In 
this  manner,  the  general  truth,  the  Second  Law  of 
Motion,  was  established  completely  and  beyond  dispute. 
But  when  this  law  had  been  proved  to  be  true  in  a 
general  sense,  with  such  accuracy  as  rude  experiments, 
like  those  of  Galileo  and  Gassendi,  would  admit,  it  still 
remained  to  be  ascertained  (supposing  our  knowledge  of 
the  law  to  be  the  result  of  experience  alone,)  whether  it 
were  true  with  that  precise  and  rigorous  exactness  which 
more  refined  modes  of  experimenting  could  test.  We 
so  willingly  believe  in  the  simplicity  of  laws  of  nature, 
that  the  rigorous  accuracy  of  such  a  law,  known  to  be  at 
least  approximately  true,  was  taken  for  granted,  till  some 
ground  for  suspecting  the  contrary  should  appear.  Yet 
calculations  have  not  been  wanting  which  might  confirm 
the  law  as  true  to  the  last  degree  of  accuracy.  Laplace 
relates  (Syst.  du  Monde,  livre  iv.,  chap.  1C,)  that  at  one 
time  he  had  conceived  it  possible  that  the  effect  of 
gravity  upon  the  moon  might  be  slightly  modified  by  the 
moon's  direction  and  velocity ;  and  that  in  this  way  an 
explanation  might  be  found  for  the  moon's  deceleration 
(a  deviation  of  her  observed  from  her  calculated  place, 
which  long  perplexed  mathematicians).  But  it  was  after 
some  time  discovered  that  this  feature  in  the  moon's 
motion  arose  from  another  cause;  and  the  second  law  of 

*    I)id/o<fu,  n.  ji.  11-1. 


motion   was  confirmed   as   true    in    the   most    rigorous 

Thus  we  see  that  although  there  were  arguments 
which  might  be  urged  in  favour  of  this  law,  founded 
upon  the  necessary  relations  of  ideas,  men  became  con 
vinced  of  its  truth  only  when  it  was  verified  and  con 
firmed  by  actual  experiment.  But  yet  in  this  case 
again,  as  in  the  former  ones,  when  the  law  had  been 
established  beyond  doubt  or  question,  men  were  very 
ready  to  believe  that  it  was  not  a  mere  result  of  observa 
tion, — that  the  truth  which  it  contained  was  not  derived 
from  experience, — that  it  might  have  been  assumed  as 
true  in  virtue  of  reasonings  anterior  to  experience, — and 
that  experiments  served  only  to  make  the  law  more  plain 
and  intelligible,  as  visible  diagrams  in  geometry  serve  to 
illustrate  geometrical  truths ;  our  knowledge  not  being 
(they  deemed)  in  mechanics,  any  more  than  in  geometry, 
borrowed  from  the  senses.  It  was  thought  by  many  to 
be  self-evident,  that  the  effect  of  a  force  in  any  direction 
cannot  be  increased  or  diminished  by  any  motion  trans 
verse  to  the  direction  of  the  force  which  the  body  may 
have  at  the  same  time :  or,  to  express  it  otherwise,  that 
if  the  motion  of  the  body  be  compounded  of  a  horizontal 
and  vertical  motion,  the  vertical  motion  alone  will  be 
affected  by  the  vertical  force.  This  principle,  indeed, 
not  only  has  appeared  evident  to  many  persons,  but  even 
at  the  present  day  is  assumed  as  an  axiom  by  many  of 
the  most  eminent  mathematicians.  It  is,  for  example, 
so  employed  in  the  Mccanique  Celeste  of  Laplace,  which 
may  be  looked  upon  as  the  standard  of  mathematical 
mechanics  in  our  time;  and  in  the  Mecanique  Analy- 
tique  of  Lagrange,  the  most  consummate  example  which 
has  appeared  of  subtilty  of  thought  on  such  subjects,  as 
well  as  of  power  of  mathematical  generalization*.  And 

*  I  may  observe  that  the  rule  that  we  may  compound  motions,  as 


thus  we  have  here  another  example  of  that  circumstance 
which  we  have  already  noticed  in  speaking  of  the  First 
Law  of  Motion,  (Art.  2  of  this  Chapter,)  and  of  the  Law 
that  Gravity  is  a  uniform  Force,  (Art.  3) ;  namely,  that 
the  law,  though  historically  established  by  experiments, 
appears,  when  once  discovered  and  reduced  to  its  most 
simple  and  general  form,  to  be  self-evident.  I  am  the 
more  desirous  of  drawing  attention  to  this  feature  in 
various  portions  of  the  history  of  science,  inasmuch  as  it 
will  be  found  to  lead  to  some  very  extensive  and  impor 
tant  views,  hereafter  to  be  considered. 

5.  The  Third  Law  of  Motion. — We  have,  in  the 
definition  of  Accelerating  Force,  a  measure  of  Forces,  so 
far  as  they  are  concerned  in  producing  motion.  We  had 
before,  in  speaking  of  the  principles  of  statics,  defined 
the  measure  of  Forces  or  Pressures,  so  far  as  they  are 
employed  in  producing  equilibrium.  But  these  two 
aspects  of  Force  are  closely  connected;  and  we  require  a 
law  which  shall  lay  down  the  rule  of  their  connexion. 
By  the  same  kind  of  muscular  exertion  by  which  we 

the  Law  supposes,  is  involved  in  the  step  of  resolving  them  ;  which  is 
done  in  the  passage  to  which  I  refer  (Mcc.  Analyt.  Ptie.  I.,  sect.  i.  art.  3, 
p.  225).  "  Si  on  concoit  que  la  mouvement  d'un  corps  et  les  forces 
qui  le  sollicitent  soient  decomposes  suivant  trois  lignes  droites  perpen- 
diculaires  entre  elles,  on  pourra  considerer  separement  les  mouvemens 
et  les  forces  relatives  a  rhacun  a  de  ces  trois  directions.  Car  a  cause  de 
la  pi-rpendicularite  dcs  directions  il  est  visible  que  chacun  de  ces  mouve- 
meiis  partiels  peut  etre  regarde  comme  independant  des  deux  autres, 
et  qu'il  ne  peut  recevoir  d'alteration  que  de  la  part  de  la  force  qui  agit 
dans  la  direction  de  ce  mouvement ;  Ton  pent  conclure  que  ces  trois 
inouvements  doivent  suivre,  chacun  en  particulier,  les  lois  des  mouve- 
tiiriis  rectilignes  acceleres  ou  retardes  par  les  forces  donnees."  Laplace 
makes  the  same  assumption  in  effect,  (Mcc  Cel.  P.  I.,  liv.  i.,  art.  7») 
hy  resolving  the  forces  which  act  upon  a  point  in  three  rectangular 
din-rtimis,  and  reasoning  separately  concerning  each  direction.  But  iu 
lii>  mode  of  treating  the  subject  is  involved  a  principle  which  belongs 
to  the  Third  Law  of  Motion,  namely,  the  doctrine  that  the  velocity  is 
;i-  the  force,  of  which  we  shall  have  to  speak  elsewhere. 


can  support  a  heavy  stone,  we  can  also  put  it  in  motion. 
The  question  then  occurs,  how  is  the  rate  and  manner 
of  its  motion  determined  ?  The  answer  to  this  question 
is  contained  in  the  Third  Law  of  Motion,  and  it  is  to 
this  effect :  that  the  Momentum  which  any  pressure  pro 
duces  in  the  mass  in  a  given  time  is  proportional  to  the 
pressure.  By  Momentum  is  meant  the  product  of  the 
numbers  which  express  the  velocity  and  the  mass  of  the 
body :  and  hence,  if  the  mass  of  the  body  be  the  same 
in  the  instances  which  we  compare,  the  rule  is, — That 
the  Telocity  is  as  the  force  which  produces  it;  and  this  is 
one  of  the  simplest  ways  of  expressing  the  Third  Law 
of  Motion. 

In  agreement  with  our  general  plan,  we  have  to  ask. 
What  is  the  ground  of  this  rule  ?  What  is  the  simplest 
and  most  satisfactory  form  to  which  we  can  reduce  the 
proof  of  it  ?  Or,  to  take  an  instance ;  if  a  double  pres 
sure  be  exerted  against  a  given  mass,  so  disposed  as  to 
be  capable  of  motion,  why  must  it  produce  twice  the 
velocity  in  the  same  time  ? 

To  answer  this  question,  suppose  the  double  pressure 
to  be  resolved  into  two  single  pressures :  one  of  these 
will  produce  a  certain  velocity;  and  the  question  is,  why 
an  equal  pressure,  acting  upon  the  same  mass,  will  pro 
duce  an  equal  velocity  in  addition  to  the  former?  Or, 
stating  the  matter  otherwise,  the  question  is,  why  each 
of  the  two  forces  will  produce  its  separate  effect,  unal 
tered  by  the  simultaneous  action  of  the  other  force  ? 

This  statement  of  the  case  makes  it  seem  to  approach 
very  near  to  such  cases  as  are  included  in  the  Second 
Law  of  Motion,  and  therefore  it  might  appear  that  this 
Third  Law  has  no  grounds  distinct  from  the  Second. 
But  it  must  be  recollected  that  the  \\ordforce  has  a  dif 
ferent  meaning  in  this  case  and  in  that ;  in  this  place  it 
signifies  pressure;  in  the  statement  of  the  Second  Law 


its  import  was  accelerative  or  deflective  force,  measured 
by  the  velocity  or  deflexion  generated.  And  thus  the 
Third  Law  of  Motion,  so  far  as  our  reasonings  yet  go, 
appears  to  rest  on  a  foundation  different  from  the  Second. 

Accordingly,  that  part  of  tho  Third  Law  of  Motion 
which  we  are  now  considering,  that  the  velocity  gene 
rated  is  as  the  force,  was  obtained,  in  fact,  by  a  separate 
train  of  research.  The  first  exemplification  of  this  law 
which  was  studied  by  mathematicians,  was  the  motion 
of  bodies  upon  inclined  planes :  for  the  force  which  urges 
a  body  down  an  inclined  plane  is  known  by  statics,  and 
hence  the  velocity  of  its  descent  was  to  be  determined. 
Galileo  originally*  in  his  attempts  to  solve  this  problem 
of  the  descent  of  a  body  down  an  inclined  plane,  did  not 
proceed  from  the  principle  which  we  have  stated,  (the 
determination  of  the  force  which  acts  down  the  inclined 
plane  from  statical  considerations,)  obvious  as  it  may 
seem ;  but  assumed,  as  we  have  already  seen,  a  propo 
sition  apparently  far  more  precarious ; — namely,  that 
a  body  sliding  down  a  smooth  inclined  plane  acquires 
always  the  same  velocity,  so  long  as  the  vertical  height 
fallen  through  is  the  same.  And  this  conjecture,  (for 
at  first  it  was  nothing  more  than  a  conjecture,)  he 
confirmed  by  an  ingenious  experiment ;  in  which  bodies 
acquired  or  lost  the  same  velocity  by  descending  or 
ascending  through  the  same  height,  although  their  paths 
were  different  in  other  respects. 

This  was  the  form  in  which  the  doctrine  of  the  mo 
tion  of  bodies  down  inclined  planes  was  at  first  presented 
in  Galileo's  Dialogues  on  the  Science  of  Motion.  But 
his  disciple  Viviani  was  dissatisfied  with  the  assumption 
thus  introduced  ;  and  in  succeeding  editions  of  the  Dia 
logues,  the  apparent  chasm  in  the  reasoning  was  much 
narrowed,  by  making  the  proof  depend  upon  a  principle 

*  /)/'«/.  delta  Sc.  NHOI<.  in.,  p. '.)().  Sec  Hist.  Ind.  Sci.  R  vi.  c.  ii.  (sect.  5. 


nearly  identical  with  the  third  law  of  motion  as  we  have 
just  stated  it.  In  the  proof  thus  added,  "  We  are  agreed," 
says  the  interlocutor"",  "that  in  a  moving  body  the 
impetus,  energy,  momentum,  or  propension  to  motion,  is 
as  great  as  is  the  force  or  least  resistance  which  suffices 
to  sustain  it ;"  and  the  impetus  or  momentum,  in  the 
course  of  the  proof,  being  taken  to  be  as  the  velocity 
produced  in  a  given  time,  it  is  manifest  that  the  prin 
ciple  so  stated  amounts  to  this ;  that  the  velocity  pro 
duced  is  as  the  statical  force.  And  thus  this  law  of 
motion  appears,  in  the  school  of  Galileo,  to  have  been 
suggested  and  established  at  first  by  experiment,  but 
afterwards  confirmed  and  demonstrated  by  a  priori 

We  see,  in  the  above  reasoning,  a  number  of  abstract 
terms  introduced  which  are  not,  at  first  at  least,  very 
distinctly  defined,  as  impetus,  momentum,  &c.  Of 
these,  momentum  has  been  selected,  to  express  that 
quantity  which,  in  a  moving  body,  measures  the  statical 
force  impressed  upon  the  body.  This  quantity  is,  as  we 
have  just  seen,  proportional  to  the  velocity  in  a  given 
body.  It  is  also,  in  different  bodies,  proportional  to  the 
mass  of  the  body.  This  part  of  the  third  law  of  motion 
follows  from  our  conception  of  matter  in  general  as  con 
sisting  of  parts  capable  of  addition.  A  double  pressure 
must  be  required  to  produce  the  same  velocity  in  a 
double  mass ;  for  if  the  mass  be  halved,  each  half  will 
require  an  equal  pressure ;  and  the  addition,  both  of  the 
pressures  and  of  the  masses,  will  take  place  without  dis 
turbing  the  effects. 

The  measure  of  the  quantity  of  matter  of  a  body  con 
sidered  as  affecting  the  velocity  which  pressure  produces 
in  the  body,  is  termed  its  inertia,  as  we  have  already 
stated,  (p.  190.)  Inertia  is  the  property  by  which  a 

*  Diafago,  p.  104. 


large  mass  of  matter  requires  a  greater  force  than  a 
small  mass,  to  give  it  an  equal  velocity.  It  belongs  to 
each  portion  of  matter;  and  portions  of  inertia  are 
added  whenever  portions  of  matter  are  added.  Hence 
inertia  is  as  the  quantity  of  matter ;  which  is  only  an 
other  way  of  expressing  this  third  law  of  motion,  so  far 
as  quantity  of  matter  is  concerned. 

But  how  do  we  know  the  quantity  of  matter  of  a 
body  ?  We  may  reply,  that  we  take  the  weight  as  the 
measure  of  the  quantity  of  matter :  but  we  may  then  be 
again  asked,  how  it  appears  that  the  weight  is  propor 
tional  to  the  inertia ;  which  it  must  be,  in  order  that  the 
quantity  of  matter  may  be  proportional  to  both  one  and 
the  other.  We  answer,  that  this  appears  to  be  true 
experimentally,  because  all  bodies  fall  with  equal  veloci 
ties  by  gravity,  when  the  known  causes  of  difference  are 
removed.  The  observations  of  falling  bodies,  indeed, 
are  not  susceptible  of  much  exactness :  but  experiments 
leading  to  the  same  result,  and  capable  of  great  precision, 
were  made  upon  pendulums  by  Newton  ;  as  he  relates  in 
his  Prinripia,  Book  m.,  prop.  6.  They  all  agreed,  he 
says,  with  perfect  accuracy  :  and  thus  the  weight  and  the 
inertia  are  proportional  in  all  cases,  and  therefore  each 
proportional  to  the  quantity  of  matter  as  measured  by 
the  other. 

The  conception  of  inertia,  as  we  have  already  seen  in 
chapter  v.,  involves  the  notion  of  action  and  reaction ; 
and  thus  the  laws  which  involve  inertia  depend  upon  the 
idea  of  mutual  causation.  The  rule,  that  the  velocity  is 
as  the  force,  depends  upon  the  principle  of  causation, 
that  the  effect  is  proportional  to  the  cause ;  the  effect 
bring  here  so  estimated  as  to  be  consistent  both  with 
the  other  laws  of  motion  and  with  experiment. 

But  here,  as  in  other  cases,  the  question  occurs 
Is  experiment  really  requisite  for  the  proof  of 


this  law  ?  If  we  look  to  authorities,  we  shall  be  not  a 
little  embarrassed  to  decide.  D'Alembert  is  against  the 
necessity  of  experimental  proof.  "  Why,"  says  he  *, 
"  should  we  have  recourse  to  this  principle  employed,  at 
the  present  day,  by  everybody,  that  the  force  is  propor 
tional  to  the  velocity?  ...  a  principle  resting  solely 
upon  this  vague  and  obscure  axiom,  that  the  effect  is 
proportional  to  the  cause.  We  shall  not  examine  here," 
he  adds,  "  if  this  principle  is  necessarily  true ;  we  shall 
only  avow  that  the  proofs  which  have  hitherto  been 
adduced  do  not  appear  to  us  unexceptionable :  nor  shall 
we,  with  some  geometers,  adopt  it  as  a  purely  contingent 
truth ;  which  would  be  to  ruin  the  certainty  of  me 
chanics,  and  to  reduce  it  to  be  nothing  more  than  an 
experimental  science.  We  shall  content  ourselves  with 
observing,"  he  proceeds,  "  that  certain  or  doubtful,  clear 
or  obscure,  it  is  useless  in  mechanics,  and  consequently 
ought  to  be  banished  from  the  science."  Though 
D'Alembert  rejects  the  third  law  of  motion  in  this  form, 
he  accepts  one  of  equivalent  import,  which  appears  to 
him  to  possess  axiomatic  certainty  ;  and  this  procedure 
is  in  consistence  with  the  course  which  he  takes,  of 
claiming  for  the  science  of  mechanics  more  than  mere 
experimental  truth.  On  the  contrary,  Laplace  considers 
this  third  law  as  established  by  experiment.  "  Is  the 
force,"  he  saysf,  "proportioned  to  the  velocity?  This," 
he  replies,  "  we  cannot  know  a  priori,  seeing  that  we 
are  in  ignorance  of  the  nature  of  moving  force :  we  must 
therefore,  for  this  purpose,  recur  to  experience  ;  for  all 
which  is  not  a  necessary  consequence  of  the  few  data  we 
have  respecting  the  nature  of  things,  is,  for  us,  only  a  re 
sult  of  observation."  And  again  he  saysj,  "Here,  then, 
we  have  two  laws  of  motion, — the  law  of  inertia  [the  first 
law  of  motion],  and  the  law  of  the  force  proportional  to 

*   Di/na»iique,  Fref.  p.  x.  t   Mec   Cel.  p.  15.  |.   P.  18. 


the  velocity, — which  are  given  by  observation.  They 
are  the  most  natural  and  the  most  simple  laws  which  we 
can  imagine,  and  without  doubt  they  flow  from  the  very 
nature  of  matter ;  but  this  nature  being  unknown,  they 
are,  for  us,  only  observed  facts :  the  only  ones,  however, 
which  mechanics  borrows  from  experience." 

It  will  appear,  I  think,  from  the  views  given  in  this 
and  several  other  parts  of  the  present  work,  that  we  can 
not  with  justice  say  that  we  have  very  "few  data  respect 
ing  the  nature  of  things,"  in  speculating  concerning  the 
laws  of  the  universe ;  since  all  the  consequences  which 
flow  from  the  relations  of  our  fundamental  ideas,  neces 
sarily  regulate  our  knowledge  of  things,  so  far  as  we 
have  any  such  knowledge.  Nor  can  we  say  that  the  na 
ture  of  matter  is  unknown  to  us,  in  any  sense  in  which 
we  can  conceive  knowledge  as  possible.  The  nature  ol 
matter  is  no  more  unknown  than  the  nature  of  space  or 
of  number.  In  our  conception  of  matter,  as  of  space 
and  of  number,  are  involved  certain  relations,  which  are 
the  necessary  groundwork  of  our  knowledge ;  and  any 
thing  which  is  independent  of  these  relations,  is  not  un 
known,  but  inconceivable. 

It  must  be  already  clear  to  the  reader,  from  the 
phraseology  employed  by  these  two  eminent  mathema 
ticians,  that  the  question  respecting  the  formation  of  the 
third  law  of  motion  can  only  be  solved  by  a  careful  con 
sideration  of  what  we  mean  by  observation  and  experi 
ence,  nature  and  matter.  But  it  will  probably  be  gene 
rally  allowed,  that,  taking  into  account  the  explanations 
already  offered  of  the  necessary  conditions  of  experience 
and  of  the  conception  of  inertia,  this  law  of  motion,  that 
the  inertia  is  as  the  quantity  of  matter,  is  almost  or  alto 
gether  self-evident. 

C.  Action  and  Reaction  are  Equal  in  Moving  Bodies. 
—When  we  have  to  consider  bodies  as  acting  upon  one 


another,  and  influencing  each  other's  motions,  the  third 
law  of  motion  is  still  applied ;  but  along  with  this,  we 
also  employ  the  general  principle  that  action  and  reaction 
are  equal  and  opposite.  Action  and  reaction  are  here  to 
be  understood  as  momentum  produced  and  destroyed, 
according  to  the  measure  of  action  established  by  the 
Third  Law  of  Motion :  and  the  cases  in  which  this  prin 
ciple  is  thus  employed  form  so  large  a  portion  of  those 
in  which  the  third  law  of  motion  is  used,  that  some 
writers  (Newton  at  the  head  of  them)  have  stated  the 
equality  of  action  and  reaction  as  the  third  law  of  motion. 
The  third  law  of  motion  being  once  established,  the 
equality  of  action  and  reaction,  in  the  sense  of  mo 
mentum  gained  and  lost,  necessarily  follows.  Thus,  if 
a  weight  hanging  by  a  string  over  the  edge  of  a  smooth 
level  table  draw  another  weight  along  the  table,  the 
hanging  weight  moves  more  slowly  than  it  would  do  if 
not  so  connected,  and  thus  loses  velocity  by  the  con 
nexion  ;  while  the  other  weight  gains  by  the  connexion 
all  the  velocity  which  it  has,  for  if  left  to  itself  it  would 
rest.  And  the  pressures  which  restrain  the  descent  of  the 
first  bodv  and  accelerate  the  motion  of  the  second,  are 


equal  at  all  instants  of  time,  for  each  of  these  pressures 
is  the  tension  of  the  string:  and  hence,  by  the  third  law 
of  motion,  the  momentum  gained  by  the  one  body,  and 
the  momentum  lost  by  the  other  in  virtue  of  the  action 
of  this  string,  are  equal.  And  similar  reasoning  may  be 
employed  in  any  other  case  where  bodies  are  connected. 
The  case  where  one  body  does  not  push  or  draw, 
but  strikes  another,  appeared  at  first  to  mechanical  rea- 
soners  to  be  of  a  different  nature  from  the  others ;  but  a 
little  consideration  was  sufficient  to  show  that  a  blow 
is,  in  fact,  only  a  short  and  violent  pressure ;  and  that, 
therefore,  the  general  rule  of  the  equality  of  momentum 
lost  and  gained  applies  to  this  as  well  as  to  the  other  cases. 


Thus,  in  order  to  determine  the  case  of  the  direct 
action  of  bodies  upon  one  another,  we  require  no  new 
Ia\v  of  motion.  The  equality  of  action  and  reaction, 
which  enters  necessarily  into  every  conception  of  me 
chanical  operation,  combined  with  the  measure  of  action 
as  given  by  the  third  law  of  motion,  enables  us  to  trace 
the  consequences  of  every  case,  whether  of  pressure  or 
of  impact. 

7.  D 'Alembert 's  Principle. — But  what  will  be  the 
result  when  bodies  do  not  act  directly  upon  each  other, 
but  are  indirectly  connected  in  any  way  by  levers,  strings, 
pulleys,  or  in  any  other  manner,  so  that  one  part  of  the 
system  has  a  mechanical  advantage  over  another?  The 
result  must  still  be  determined  by  the  principle  that 
action  and  reaction  balance  each  other.  The  action  and 
reaction,  being  pressures  in  one  sense,  must  balance  each 
other  by  the  laws  of  statics,  for  these  laws  determine 
the  equilibrium  of  pressure.  Now  action  and  reaction, 
according  to  their  measures  in  the  Third  Law  of  Motion, 
are  momentum  gained  and  lost,  when  the  action  is  di 
rect  ;  and  except  the  indirect  action  introduce  some 
modification  of  the  law,  they  must  have  the  same  mea 
sure  still.  But,  in  fact,  we  cannot  well  conceive  any 
modification  of  the  law  to  take  place  in  this  case;  for 
direct  action  is  only  one  (the  ultimate)  case  of  indirect 
action.  Thus  if  two  heavy  bodies  act  at  different  points 
of  a  lever,  the  action  of  each  on  the  other  is  indirect ; 
but  if  the  two  points  come  together,  the  action  becomes 
direct.  Hence  the  rule  must  be  that  which  we  have 
already  stated ;  for  if  the  rule  were  false  for  indirect 
action,  it  would  also  be  false  for  direct  action,  for  which 
case  we  have  shown  it  to  be  true.  And  thus  we  obtain 
the  general  principle,  that  in  any  system  of  bodies  which 
act  on  each  other,  action  and  reaction,  estimated  by  mo 
mentum  gained  and  lost,  balance  each  other  according 


to  the  laws  of  equilibrium.  This  principle,  which  is  so 
general  as  to  supply  a  key  to  the  solution  of  all  pos 
sible  mechanical  problems,  is  commonly  called  UAlem- 
lerfs  Principle.  The  experimental  proofs  which  con 
vinced  men  of  the  truth  of  the  Third  Law  of  Motion 
were,  many  or  most  of  them,  proofs  of  the  law  in  this 
extended  sense.  And  thus  the  proof  of  D'Alembert's 
Principle,  both  from  the  idea  of  mechanical  action  and 
from  experience,  is  included  in  the  proof  of  the  law 
already  stated. 

8.  Connexion  of  Dynamical  and  Statical  Principles. 
—The  principle  of  equilibrium  of  D'Alembert  just  stated, 
is  the  law  which  he  would  substitute  for  the  Third  Law 
of  Motion ;  and  he  would  thus  remove  the  necessity  for 
an  independent  proof  of  that  law.  In  like  manner,  the 
Second  Law  of  Motion  is  by  some  writers  derived  from 
the  principle  of  the  composition  of  statical  forces ;  and 
they  would  thus  supersede  the  necessity  of  a  reference  to 
experiment  in  that  case.  Laplace  takes  this  course,  and 
thus,  as  we  have  seen,  rests  only  the  First  and  Third  Law 
of  Motion  upon  experience.  Newton,  on  the  other  hand, 
recognizes  the  same  connexion  of  propositions,  but  for 
a  different  purpose ;  for  he  derives  the  composition  of 
statical  forces  from  the  Second  Law  of  Motion. 

The  close  connexion  of  these  three  principles,  the 
composition  of  (statical)  forces,  the  composition  of  (ac 
celerating)  forces  with  velocities,  and  the  measure  of 
(moving)  forces  by  velocities,  cannot  be  denied ;  yet  it 
appears  to  be  by  no  means  easy  to  supersede  the  neces 
sity  of  independent  proofs  of  the  two  last  of  these  prin 
ciples.  Both  may  be  proved  or  illustrated  by  expe 
riment  :  and  the  experiments  which  prove  the  one  are 
different  from  those  which  establish  the  other.  For 
example,  it  appears  by  easy  calculations,  that  when  we 
apply  our  principles  to  the  oscillations  of  a  pendulum, 


the  Second  Law  is  proved  by  the  fact,  that  the  oscilla 
tions  take  place  at  the  same  rate  in  an  east  and  west, 
and  in  a  north  and  south  direction :  under  the  same  cir 
cumstances,  the  Third  Law  is  proved  by  our  finding  that 
the  time  of  a  small  oscillation  is  proportional  to  the 
square  root  of  the  length  of  a  pendulum ;  and  similar 
differences  might  be  pointed  out  in  other  experiments, 
as  to  their  bearing  upon  the  one  law  or  the  other. 

9.  Mechanical  Principles  become  gradually  more 
simple  and  more  evident. — I  will  again  point  out  in 
general  two  circumstances  which  I  have  already  noticed 
in  particular  cases  of  the  laws  of  motion. — Truths  are 
often  at  first  assumed  in  a  form  which  is  far  from  being 
the  most  obvious  or  simple ; — and  truths  once  discovered 
are  gradually  simplified,  so  as  to  assume  the  appearance 
of  self-evident  truths. 

The  former  circumstance  is  exemplified  in  several  of 
the  instances  which  we  have  had  to  consider.  The 
assumption  that  a  perpetual  motion  is  impossible  pre 
ceded  the  knowledge  of  the  first  law  of  motion.  The 
assumed  equality  of  the  velocities  acquired  down  two  in 
clined  planes  of  the  same  height,  was  afterwards  reduced 
to  the  third  law  of  motion  by  Galileo  himself.  In  the 
History  *,  we  have  noted  Huyghens's  assumption  of  the 
equality  of  the  actual  descent  and  potential  ascent  of  the 
center  of  gravity :  this  was  afterwards  reduced  by  Her 
man  and  the  Bernoullis,  to  the  statical  equivalence  of  the 
solicitations  of  gravity  and  the  vicarious  solicitations  of 
the  effective  forces  which  act  on  each  point ;  and  finally 
to  the  principle  of  D'Alembert,  which  asserts  that  the 
motions  gained  and  lost  balance  each  other. 

This  assertion  of  principles  which  now  appear  neither 
obvious  nor  self-evident,  is  not  to  be  considered  as  a 
groundless  assumption  on  the  part  of  the  discoverers  by 

*   B.  vi.  c.  v.  sect.  2. 
VOL.  I.    W.  P.  R 


whom  it  was  made.  On  the  contrary,  it  is  evidence  of 
the  deep  sagacity  and  clear  thought  which  were  requisite 
in  order  to  make  such  discoveries.  For  these  results  are 
really  rigorous  consequences  of  the  laws  of  motion  in 
their  simplest  form :  and  the  evidence  of  them  was  pro 
bably  present,  though  undeveloped,  in  the  minds  of  the 
discoverers.  We  are  told  of  geometrical  students,  who, 
by  a  peculiar  aptitude  of  mind,  perceived  the  evidence  of 
some  of  the  more  advanced  propositions  of  geometry 
without  going  through  the  introductory  steps.  We  must 
suppose  a  similar  aptitude  for  mechanical  reasonings, 
which,  existing  in  the  minds  of  Stevinus,  Galileo,  New 
ton,  and  Huyghens,  led  them  to  make  those  assumptions 
which  finally  resolved  themselves  into  the  laws  of  motion. 
We  may  observe  further,  that  the  simplicity  and  evi 
dence  which  the  laws  of  mechanics  have  at  length  as 
sumed,  are  much  favoured  by  the  usage  of  words  among 
the  best  writers  on  such  subjects.  Terms  which  origi 
nally,  and  before  the  laws  of  motion  were  fully  known, 
were  used  in  a  very  vague  and  fluctuating  sense,  were 
afterwards  limited  and  rendered  precise,  so  that  asser 
tions  which  at  first  appear  identical  propositions  become 
distinct  and  important  principles.  Thus  force,  motion, 
momentum,  are  terms  which  were  employed,  though  in  a 
loose  manner,  from  the  very  outset  of  mechanical  specu 
lation.  And  so  long  as  these  words  retained  the  vagueness 
of  common  language,  it  would  have  been  a  useless  and 
barren  truism  to  say  that  "  the  momentum  is  proportional 
to  the  force,"  or  that  "a  body  loses  as  much  motion  as 
it  communicates  to  another."  But  when  "  momentum  " 
and  "quantity  of  motion"  are  defined  to  mean  the  pro 
duct  of  mass  and  velocity,  these  two  propositions  imme 
diately  become  distinct  statements  of  the  third  law  of 
motion  and  its  consequences.  In  like  manner,  the  asser 
tion  that  "gravity  is  a  uniform  force"  wras  assented  to, 


before  it  was  settled  what  a  uniform  force  was ;  but  this 
assertion  only  became  significant  and  useful  when  that 
point  had  been  properly  determined.  The  statement 
that  "when  different  motions  are  communicated  to  the 
same  body  their  effects  are  compounded,"  becomes  the 
second  law  of  motion,  when  we  define  what  composition 
of  motions  is.  And  the  same  process  may  be  observed 
in  other  cases. 

And  thus  we  see  how  well  the  form  which  science 
ultimately  assumes  is  adapted  to  simplify  knowledge. 
The  definitions  which  are  adopted,  and  the  terms  which 
become  current  in  precise  senses,  produce  a  complete 
harmony  between  the  matter  and  the  form  of  our  know 
ledge;  so  that  truths  which  were  at  first  unexpected  and 
recondite,  became  familiar  phrases,  and  after  a  few  gene 
rations  sound,  even  to  common  ears,  like  identical  pro 

10.  Controversy  of  the  Measure  of  Force. — In  the 
History  of  Mechanics'*"",  we  have  given  an  account  of  the 
controversy  which,  for  some  time,  occupied  the  mathema 
ticians  of  Europe,  whether  the  forces  of  bodies  in  motion 
should  be  reckoned  proportional  to  the  velocity,  or  to  the 
square  of  the  velocity.  We  need  not  here  recall  the 
events  of  this  dispute ;  but  we  may  remark,  that  its  his 
tory,  as  a  metaphysical  controversy,  is  remarkable  in  this 
respect,  that  it  has  been  finally  and  completely  settled; 
for  it  is  now  agreed  among  mathematicians  that  both 
sides  were  right,  and  that  the  results  of  mechanical  action 
may  be  expressed  with  equal  correctness  by  means  of 
momentum  and  of  vis  viva.  It  is,  in  one  sense,  as  D'Alem- 
bert  has  saidf,  a  dispute  about  words;  but  we  are  not 

*  B.  vi.  c.  v.  sect.  2. 

t  D'Alcinbcrt  has  also  remarked  (Dynatniqtie,  Pref.  xxi.,)  that  this 
controversy  "  shows  how  little  justice  and  precision  there  is  in  the 
pretended  axiom  that  causes  are  proportional  to  their  effects."  But 



to  infer  that,  on  that  account,  it  was  frivolous  or  useless ; 
for  such  disputes  are  one  principal  means  of  reducing  the 
principles  of  our  knowledge  to  their  utmost  simplicity 
and  clearness.  The  terms  which  are  employed  in  the 
science  of  mechanics  are  now  liberated  for  ever,  in  the 
minds  of  mathematicians,  from  that  ambiguity  which 
was  the  battle-ground  in  the  war  of  the  vis  viva. 

But  we  may  observe  that  the  real  reason  of  this  con 
troversy  was  exactly  that  tendency  which  we  have  been 
noticing ; — the  disposition  of  man  to  assume  in  his  specu 
lations  certain  general  propositions  as  true,  and  to  fix  the 
sense  of  terms  so  that  they  shall  fall  in  with  this  truth. 
It  was  agreed,  on  all  hands,  that  in  the  mutual  action  of 
bodies  the  same  quantity  of  force  is  always  preserved; 
and  the  question  was,  by  which  of  the  two  measures  this 
rule  could  best  be  verified.  We  see,  therefore,  that  the 
dispute  was  not  concerning  a  definition  merely,  but  con 
cerning  a  definition  combined  with  a  general  proposition. 
Such  a  question  may  be  readily  conceived  to  have  been 
by  no  means  unimportant ;  and  we  may  remark,  in  pass 
ing,  that  such  controversies,  although  they  are  commonly 
afterwards  stigmatized  as  quarrels  about  words  and  defi 
nitions,  are,  in  reality,  events  of  considerable  conse 
quence  in  the  history  of  science  ;  since  they  dissipate  all 
ambiguity  and  vagueness  in  the  use  of  terms,  and  bring 
into  view  the  conditions  under  which  the  fundamental 
principles  of  our  knowledge  can  be  most  clearly  and 
simply  presented. 

It  is  worth  our  while  to  pause  for  a  moment  on  the 
prospect  that  we  have  thus  obtained,  of  the  advance  of 

this  reflection  is  by  no  means  well  founded.  For  since  both  measures 
are  true,  it  appears  that  causes  may  be  justly  measured  by  their  effects, 
even  when  very  different  kinds  of  effects  are  taken.  That  the  axiom 
does  not  point  out  one  precise  measure,  till  illustrated  by  experience  or  j 
by  other  considerations,  we  grant  :  but  the  same  thing  occurs  in  the 
application  of  other  axioms  also. 


knowledge,  as  exemplified  in  the  history  of  Mechanics. 
The  general  transformation  of  our  views  from  vague  to 
definite,  from  complex  to  simple,  from  unexpected  dis 
coveries  to  self-evident  truths,  from  seeming  contradic 
tions  to  identical  propositions,  is  very  remarkable,  but  it 
is  by  no  means  peculiar  to  our  subject.  The  same  cir 
cumstances,  more  or  less  prominent,  more  or  less  deve 
loped,  appear  in  the  history  of  other  sciences,  according 
to  the  point  of  advance  which  each  has  reached.  They 
bear  upon  very  important  doctrines  respecting  the  pro 
spects,  the  limits,  and  the  very  nature  of  our  knowledge. 
And  though  these  doctrines  require  to  be  considered  with 
reference  to  the  whole  body  of  science,  yet  the  peculiar 
manner  in  which  they  are  illustrated  by  the  survey  of 
the  history  of  Mechanics,  on  which  we  have  just  been 
engaged,  appears  to  make  this  a  convenient  place  for 
introducing  them  to  the  reader. 



1.  IT  was  formerly  stated"-  that  experience  cannot 
establish  any  universal  or  necessary  truths.  The  number 
of  trials  which  we  can  make  of  any  proposition  is  neces 
sarily  limited,  and  observation  alone  cannot  give  us  any 
ground  of  extending  the  inference  to  untried  cases.  Ob 
served  facts  have  no  visible  bond  of  necessary  connexion, 
and  no  exercise  of  our  senses  can  enable  us  to  discover 
such  connexion.  We  can  never  acquire  from  a  mere 
observation  of  facts,  the  right  to  assert  that  a  proposition 
is  true  in  all  cases,  and  that  it  could  not  be  otherwise 
than  we  find  it  to  be. 

*  B.  i.,  c.  v.     Of  Experience. 


Yet,  as  we  have  just  seen  in  the  history  of  the  laws  of 
motion,   we  may  go  on  collecting  our  knowledge  from 
observation,  and  enlarging  and  simplifying  it,  till  it  ap 
proaches  or  attains  to  complete  universality  and  seeming 
necessity.    Whether  the  laws  of  motion,  as  we  now  know 
them,  can  be  rigorously  traced  to  an  absolute  necessity  in 
the  nature  of  things,  we  have  not  ventured  absolutely  to 
pronounce.     But  we  have  seen  that  some  of  the  most 
acute  and  profound  mathematicians  have  believed  that, 
for  these  laws  of  motion,  or  some  of  them,  there  was 
such  a  demonstrable  necessity  compelling  them  to  be 
such  as  they  are,  and  no  other.    Most  of  those  who  have 
carefully  studied  the  principles  of  Mechanics  will  allow 
that  some  at  least  of  the  primary  laws  of  motion  approach 
very  near  to  this  character  of  necessary  truth ;  and  will 
confess  that  it  would  be  difficult  to  imagine  any  other 
consistent  scheme  of  fundamental  principles.    And  almost 
all  mathematicians  will  allow  to  these  lawrs  an  absolute 
universality  ;  so  that  we  may  apply  them  without  scruple 
or  misgiving,  in  cases  the  most  remote  from  those  to 
which  our  experience  has  extended.     What  astronomer 
would  fear  to  refer  to  the  known  laws  of  motion,  in  rea 
soning  concerning  the  double  stars;  although  these  objects 
are  at  an  immeasurably  remote  distance  from  that  solar 
system  which  has  been  the  only  field  of  our  observation 
of  mechanical  facts?    What  philosopher,  in  speculating 
respecting  a  magnetic  fluid,  or  a  luminiferous  ether,  would 
hesitate  to  apply  to  it  the  mechanical  principles  which 
are  applicable  to  fluids  of  known  mechanical  properties  ? 
When  we  assert  that  the  quantity  of  motion  in  the  world 
cannot  be  increased  or  diminished  by  the  mutual  actions 
of  bodies,  does  not  every  mathematician  feel  convinced 
that  it  would  be  an  unphilosophical  restriction  to  limit 
this  proposition  to  such   modes  of  action  as  we  have 


Yet  no  one  can  doubt  that,  in  historical  fact,  these 
laws  were  collected  from  experience.  That  such  is  the 
case,  is  no  matter  of  conjecture.  We  know  the  time,  the 
persons,  the  circumstances,  belonging  to  each  step  of  each 
discovery.  I  have,  in  the  History,  given  an  account  of 
these  discoveries ;  and  in  the  previous  chapters  of  the  pre 
sent  work,  I  have  further  examined  the  nature  and  the 
import  of  the  principles  which  were  thus  brought  to  light. 

Here,  then,  is  an  apparent  contradiction.  Experi 
ence,  it  would  seem,  has  done  that  which  we  had  proved 
that  she  cannot  do.  She  has  led  men  to  propositions, 
universal  at  least,  and  to  principles  which  appear  to  some 
persons  necessary.  What  is  the  explanation  of  this  con 
tradiction,  the  solution  of  this  paradox  ?  Is  it  true  that 
Experience  can  reveal  to  us  universal  and  necessary 
truths  ?  Does  she  possess  some  secret  virtue,  some  un 
suspected  power,  by  which  she  can  detect  connexions 
and  consequences  which  we  have  declared  to  be  out  of 
her  sphere?  Can  she  see  more  than  mere  appearances, 
and  observe  more  than  mere  facts  ?  Can  she  penetrate, 
in  some  way,  to  the  nature  of  things  ? — descend  below  the 
surface  of  phenomena  to  their  causes  and  origins,  so  as 
to  be  able  to  say  what  can  and  what  can  not  be  ;• — what 
occurrences  are  partial,  and  what  universal  ?  If  this  be 
so,  we  have  indeed  mistaken  her  character  and  powers ; 
and  the  whole  course  of  our  reasoning  becomes  pre 
carious  and  obscure.  But,  then,  when  we  return  upon 
our  path  we  cannot  find  the  point  at  which  we  deviated, 
we  cannot  detect  the  false  step  in  our  deduction.  It 
still  seems  that  by  experience,  strictly  so  called,  we 
cannot  discover  necessary  and  universal  truths.  Our 
senses  can  give  us  no  evidence  of  a  necessary  connexion 
in  phenomena.  Our  observation  must  be  limited,  and 
cannot  testify  concerning  anything  which  is  beyond  its 
limits.  A  general  view  of  our  faculties  appears  to  prove 


it  to  be  impossible  that  men  should  do  what  the  history 
of  the  science  of  mechanics  shows  that  they  have  done. 

2.  But  in  order  to  try  to  solve  this  Paradox,  let  us 
again  refer  to  the  History  of  Mechanics.  In  the  cases 
belonging  to  that  science,  in  which  propositions  of  the 
most  unquestionable  universality,  and  most  approaching 
to  the  character  of  necessary  truths,  (as,  for  instance,  the 
laws  of  motion,)  have  been  arrived  at,  what  is  the  source 
of  the  axiomatic  character  which  the  propositions  thus 
assume  ?  The  answer  to  this  question  will,  we  may  hope, 
throw  some  light  on  the  perplexity  in  which  we  appear 
to  be  involved. 

Now  the  answer  to  this  inquiry  is,  that  the  laws 
of  motion  borrow  their  axiomatic  character  from  their 
being  merely  interpretations  of  the  Axioms  of  Causation. 
Those  axioms,  being  exhibitions  of  the  Idea  of  Cause 
under  various  aspects,  are  of  the  most  rigorous  univer 
sality  and  necessity.  And  so  far  as  the  laws  of  motion 
are  exemplifications  of  those  axioms,  these  laws  must  be 
no  less  universal  and  necessary.  How  these  axioms  are 
to  be  understood ; — in  what  sense  cause  and  effect,  action 
and  reaction,  are  to  be  taken,  experience  and  observa 
tion  did,  in  fact,  teach  inquirers  on  this  subject ;  and 
without  this  teaching,  the  laws  of  motion  could  never 
have  been  distinctly  known.  If  two  forces  act  together, 
each  must  produce  its  effect,  by  the  axiom  of  causation ; 
and,  therefore,  the  effects  of  the  separate  forces  must  be 
confounded.  But  a  long  course  of  discussion  and  expe 
riment  must  instruct  men  of  what  kind  this  composition 
of  forces  is.  Again ;  action  and  reaction  must  be  equal ; 
but  much  thought  and  some  trial  were  needed  to  show 
what  action  and  reaction  are.  Those  metaphysicians  who 
enunciated  Laws  of  motion  without  reference  to  expe 
rience,  propounded  only  such  laws  as  were  vague  and 
inapplicable.  But  yet  these  persons  manifested  the 


indestructible  conviction,  belonging  to  man's  speculative 
nature,  that  there  exist  Laws  of  motion,  that  is,  uni 
versal  formula?,  connecting  the  causes  and  effects  when 
motion  takes  place.  Those  mechanicians,  again,  who, 
observed  facts  involving  equilibrium  and  motion,  and 
stated  some  narrow  rules,  without  attempting  to  ascend 
to  any  universal  and  simple  principle,  obtained  laws  no 
less  barren  and  useless  than  the  metaphysicians ;  for 
they  could  not  tell  in  what  new  cases,  or  whether  in 
any,  their  laws  would  be  verified ; — they  needed  a  more 
general  rule,  to  show  them  the  limits  of  the  rule  they 
had  discovered.  They  went  wrong  in  each  attempt  to 
solve  a  new  problem,  because  their  interpretation  of 
the  terms  of  the  axioms,  though  true,  perhaps,  in  certain 
cases,  was  not  right  in  general. 

Thus  Pappus  erred  in  attempting  to  interpret  as  a 
case  of  the  lever,  the  problem  of  supporting  a  weight 
upon  an  inclined  plane ;  thus  Aristotle  erred  in  inter 
preting  the  doctrine  that  the  weight  of  bodies  is  the 
cause  of  their  fall ;  thus  Kepler  erred  in  interpreting  the 
rule  that  the  velocity  of  bodies  depends  upon  the  force; 
thus  Bernoulli'-"  erred  in  interpreting  the  equality  of 
action  and  reaction  upon  a  lever  in  motion.  In  each 
of  these  instances,  true  doctrines,  already  established, 
(whether  by  experiment  or  otherwise,)  were  erroneously 
applied.  And  the  error  was  corrected  by  further  reflec 
tion,  which  pointed  out  that  another  mode  of  interpreta 
tion  was  requisite,  in  order  that  the  axiom  which  was 
appealed  to  in  each  case  might  retain  its  force  in  the 
most  general  sense.  And  in  the  reasonings  which  avoided 
or  corrected  such  errors,  and  which  led  to  substantial 
general  truths,  the  object  of  the  speculator  always  was 
to  give  to  the  acknowledged  maxims  which  the  Idea  of 
Cause  suggested,  such  a  signification  as  should  be  con- 

*  Hist.  Ind.  .VcJ.,  B.  vi.  c.  v.  st-ct.  2. 


sistcnt  with  their  universal  validity.  The  rule  was  not 
accepted  as  particular  at  the  outset,  and  afterwards  gene 
ralized  more  and  more  widely ;  but  from  the  very  first, 
the  universality  of  the  rule  was  assumed,  and  the  ques 
tion  was,  how  it  should  be  understood  so  as  to  be 
universally  true.  At  every  stage  of  speculation,  the  law 
was  regarded  as  a  general  law.  This  was  not  an  aspect 
which  it  gradually  acquired,  by  the  accumulating  con 
tributions  of  experience,  but  a  feature  of  its  original 
and  native  character.  What  should  happen  universally, 
experience  might  be  needed  to  show :  but  that  what 
happened  should  happen  universally,  was  implied  in  the 
nature  of  knowledge.  The  universality  of  the  laws  of 
motion  was  not  gathered  from  experience,  however  much 
the  laws  themselves  might  be  so. 

3.  Thus  we  obtain  the  solution  of  our  Paradox,  so 
far  as  the  case  before  us  is  concerned.  The  laws  of 
motion  borrow  their  form  from  the  Idea  of  Causation, 
though  their  matter  may  be  given  by  experience :  and 
hence  they  possess  a  universality  which  experience  cannot 
give.  They  are  certainly  and  universally  valid ;  and  the 
only  question  for  observation  to  decide  is,  how  they  are 
to  be  understood.  They  are  like  general  mathematical 
formula?,  which  are  known  to  be  true,  even  while  we  are 
ignorant  what  are  the  unknown  quantities  which  they 
involve.  It  must  be  allowed,  on  the  other  hand,  that  so 
long  as  these  formula?  are  not  interpreted  by  a  real 
study  of  nature,  they  are  not  only  useless  but  prejudi 
cial  ;  filling  men's  minds  with  vague  general  terms,  empty 
maxims,  and  unintelligible  abstractions,  which  they  mis 
take  for  knowledge.  Of  such  perversion  of  the  specula 
tive  propensities  of  man's  nature,  the  world  has  seen  too 
much  in  all  ages.  Yet  we  must  not,  on  that  account, 
despise  these  forms  of  truth,  since  without  them,  no 
general  knowledge  is  possible.  Without  general  terms, 


and  maxims,  and  abstractions,  we  can  have  no  science, 
no  speculation ;  hardly,  indeed,  consistent  thought  or 
the  exercise  of  reason.  The  course  of  real  knowledge  is, 
to  obtain  from  thought  and  experience  the  right  inter 
pretation  of  our  general  terms,  the  real  import  of  our 
maxims,  the  true  generalizations  which  our  abstractions 

4.  If  it  be  asked,  How  Experience  is  able  to  teach  us 
to  interpret  aright  the  general  terms  which  the  Axioms 
of  Causation  involve ; — whence  she  derives  the  light 
which  she  is  to  throw  on  these  general  notions ;  the 
answer  is  obvious ; — namely,  that  the  relations  of  causa 
tion  are  the  conditions  of  Experience; — that  the  general 
notions  are  exemplified  in  the  particular  cases  of  which 
she  takes  cognizance.  The  events  which  take  place 
about  us,  and  which  are  the  objects  of  our  observation, 
we  cannot  conceive  otherwise  than  as  subject  to  the 
laws  of  cause  and  effect.  Every  event  must  have  a 
cause; — Every  effect  must  be  determined  by  its  cause; — 
these  maxims  are  true  of  the  phenomena  which  form 
the  materials  of  our  experience.  It  is  precisely  to  them, 
that  these  truths  apply.  It  is  in  the  world  which  we 
have  before  our  eyes,  that  these  propositions  are  univer 
sally  verified ;  and  it  is  therefore  by  the  observation  of 
what  we  see,  that  we  must  learn  how  these  propositions 
are  to  be  understood.  Every  fact,  every  experiment,  is 
an  example  of  these  statements ;  and  it  is  therefore  by 
attention  to  and  familiarity  with  facts  and  experiments, 
that  we  learn  the  signification  of  the  expressions  in  which 
the  statements  are  made ;  just  as  in  any  other  case  we 
learn  the  import  of  language  by  observing  the  manner 
in  which  it  is  applied  in  known  cases.  Experience  is 
the  interpreter  of  nature ;  it  being  understood  that  she 
is  to  make  her  interpretation  in  that  comprehensive 
phraseology  which  is  the  genuine  language  of  science. 


5.  We  may  return  for  an  instant  to  the  objection, 
that  experience  cannot  give  us  general  truths,  since, 
after  any  number  of  trials  confirming  a  rule,  we  may, 
for  aught  we  can  foresee,  have  one  which  violates  the 
rule.  When  we  have  seen  a  thousand  stones  fall  to  the 
ground,  we  may  see  one  which  does  not  fall  under  the 
same  apparent  circumstances.  How  then,  it  is  asked, 
can  experience  teach  us  that  all  stones,  rigorously  speak 
ing,  will  fall  if  unsupported  ?  And  to  this  we  reply, 
that  it  is  not  true  that  we  can  conceive  one  stone  to  be 
suspended  in  the  air,  while  a  thousand  others  fall,  with 
out  believing  some  peculiar  cause  to  support  it ;  and 
that,  therefore,  such  a  supposition  forms  no  exception  to 
the  law,  that  gravity  is  a  force  by  which  all  bodies  are 
urged  downwards.  Undoubtedly  we  can  conceive  a  body, 
when  dropt  or  thrown,  to  move  in  a  line  quite  different 
from  other  bodies:  thus  a  certain  missile*  used  by  the 
natives  of  Australia,  and  lately  brought  to  this  country, 
when  thrown  from  the  hand  in  a  proper  manner,  de 
scribes  a  curve,  and  returns  to  the  place  from  whence  it 
was  thrown.  But  did  any  one,  therefore,  even  for  an 
instant  suppose  that  the  laws  of  motion  are  different  for 
this  and  for  other  bodies?  On  the  contrary,  was  not 
every  person  of  a  speculative  turn  immediately  led  to 
inquire  how  it  was  that  the  known  causes  which  modify 
motion,  the  resistance  of  the  air  and  the  other  causes, 
produced  in  this  instance  so  peculiar  an  effect  ?  And  if 
the  motion  had  been  still  more  unaccountable,  it  would 
not  have  occasioned  any  uncertainty  whether  it  were 
consistent  with  the  agency  of  gravity  and  the  laws  of 
motion.  If  a  body  suddenly  alter  its  direction,  or  move 
in  any  other  unexpected  manner,  we  never  doubt  that 
there  is  a  cause  of  the  change.  We  may  continue  quite 
ignorant  of  the  nature  of  this  cause,  but  this  ignorance 
*  Called  the  Bo-me-rang. 


never  occasions  a  moment's  doubt  that  the  cause  exists 
and  is  exactly  suited  to  the  effect.  And  thus  experience 
can  prove  or  discover  to  us  general  rules,  but  she  can 
never  prove  that  general  rules  do  not  exist.  Anomalies, 
exceptions,  unexplained  phenomena,  may  remind  us  that 
we  have  much  still  to  learn,  but  they  can  never  make 
us  suppose  that  truths  are  not  universal.  We  may  ob 
serve  facts  that  show  us  we  have  not  fully  understood 
the  meaning  of  our  general  laws,  but  we  can  never  find 
facts  which  show  our  laws  to  have  no  meaning.  Our 
experience  is  bound  in  by  the  limits  of  cause  and  effect, 
and  can  give  us  no  information  concerning  any  region 
where  that  relation  does  not  prevail.  The  whole  series 
of  external  occurrences  and  objects,  through  all  time 
and  space,  exists  only,  and  is  conceived  only,  as  subject 
to  this  relation ;  and  therefore  we  endeavour  in  vain  to 
imagine  to  ourselves  when  and  where  and  how  excep 
tions  to  this  relation  may  occur.  The  assumption  of  the 
connexion  of  cause  and  effect  is  essential  to  our  expe 
rience,  as  the  recognition  of  the  maxims  which  express 
this  connexion  is  essential  to  our  knowledge. 

6.  I  have  thus  endeavoured  to  explain  in  some 
measure  how,  at  least  in  the  field  of  our  mechanical 
knowledge,  experience  can  discover  universal  truths, 
though  she  cannot  give  them  their  universality ;  and 
how  such  truths,  though  borrowing  their  form  from  our 
ideas,  cannot  be  understood  except  by  the  actual  study 
of  external  nature.  And  thus  with  regard  to  the  laws 
of  motion,  and  other  fundamental  principles  of  Mechanics, 
the  analysis  of  our  ideas  and  the  history  of  the  progress 
of  the  science  well  illustrate  each  other. 

If  the  paradox  of  the  discovery  of  universal  truths 
by  experience  be  thus  solved  in  one  instance,  a  much 
wider  question  offers  itself  to  us ; — How  far  the  difficulty, 
and  how  far  the  solution,  are  applicable  to  other  sub- 


jects.  It  is  easy  to  see  that  this  question  involves  most 
grave  and  extensive  doctrines  with  regard  to  the  whole 
compass  of  human  knowledge :  and  the  views  to  which 
we  have  been  led  in  the  present  Book  of  this  work  are, 
we  trust,  fitted  to  throw  much  light  upon  the  general 
aspect  of  the  subject.  But  after  discussions  so  abstract, 
and  perhaps  obscure,  as  those  in  which  we  have  been 
engaged  for  some  chapters,  I  willingly  postpone  to  a 
future  occasion  an  investigation  which  may  perhaps 
appear  to  most  readers  more  recondite  and  difficult 
still.  And  we  have,  in  fact,  many  other  special  fields 
of  knowledge  to  survey,  before  we  are  led  by  the  order 
of  our  subject,  to  those  general  questions  and  doctrines, 
those  antitheses  brought  into  view  and  again  resolved, 
which  a  view  of  the  whole  territory  of  human  know 
ledge  suggests,  and  by  which  the  nature  and  conditions 
of  knowledge  are  exhibited. 

Before  we  quit  the  subject  of  mechanical  science  we 
shall  make  a  few  remarks  on  another  doctrine  which 
forms  part  of  the  established  truths  of  the  science, 
namely,  the  doctrine  of  universal  gravitation. 



THE  doctrine  of  universal  gravitation  is  a  feature  of  so 
much  importance  in  the  history  of  science  that  we  shall 
not  pass  it  by  without  a  few  remarks  on  the  nature  and 
evidence  of  the  doctrine. 

1.  To  a  certain  extent  the  doctrine  of  the  attraction 
of  bodies  according  to  the  law  of  the  inverse  square  of 
the  distance,  exhibits  in  its  progress  among  men  the 


same  general  features  which  we  have  noticed  in  the  his 
tory  of  the  laws  of  motion.  This  doctrine  was  main 
tained  a  priori  on  the  ground  of  its  simplicity,  and  as 
serted  positively,  even  before  it  was  clearly  understood : 
— notwithstanding  this  anticipation,  its  establishment 
on  the  ground  of  facts  was  a  task  of  vast  labour  and 
sagacity  : — when  it  had  been  so  established  in  a  general 
way,  there  occurred  at  later  periods,  an  occasional  sus 
picion  that  it  might  be  approximately  true  only  : — these 
suspicions  led  to  further  researches,  which  showed  the 
rule  to  be  rigorously  exact : — and  at  present  there  are 
mathematicians  who  maintain,  not  only  that  it  is  true, 
but  that  it  is  a  necessary  property  of  matter.  A  very 
few  words  on  each  of  these  points  will  suffice. 

2.  I  have  shown  in  the  History  of  Science*,  that  the 
attraction  of  the  sun  according  to  the  inverse  square  of 
the  distance,  had  been  divined  by  Bullialdus,  Hooke, 
Halley,  and  others,  before  it  was  proved  by  Newton. 
Probably  the  reason  which  suggested  this  conjecture  was, 
that  gravity  might  be  considered  as  a  sort  of  emanation ; 
and  that  thus,  like  light  or  any  other  effect  diffused  from 
a  center,  it  must  follow  the  law  just  stated,  the  efficacy 
of  the  force  being  weakened  in  receding  from  the  center, 
exactly  in  proportion  to  the  space  through  which  it  is 
diffused.  It  cannot  be  denied  that  such  a  view  appears 
to  be  strongly  recommended  by  analogy. 

When  it  had  been  proved  by  Newton  that  the  planets 
were  really  retained  in  thein  elliptical  orbits  by  a  central 
force,  his  calculations  also  showed  that  the  above-stated 
law  of  the  force  must  be  at  least  very  approximately 
correct,  since  otherwise  the  aphelia  of  the  orbits  could 
not  be  so  nearly  at  rest  as  they  were.  Yet  when  it 
seemed  as  if  the  motion  of  the  moon's  apogee  could  not 
be  accounted  for  without  some  new  supposition,  the  d 

*  B.  VTI.  c.  i. 


priori  argument  in  favour  of  the  inverse  square  did  not 
prevent  Clairaut  from  trying  the  hypothesis  of  a  small 
term  added  to  that  which  expressed  the  ancient  law : 
but  when,  in  order  to  test  the  accuracy  of  this  hypothe 
sis,  the  calculation  of  the  motion  of  the  moon's  apogee 
was  pushed  to  a  greater  degree  of  exactness  than  had 
been  obtained  before,  it  was  found  that  the  new  term 
vanished  of  itself;  and  that  the  inverse  square  now  ac 
counted  for  the  whole  of  the  motion.  And  thus,  as  in 
the  case  of  the  second  law  of  motion,  the  most  scrupulous 
examination  terminated  in  showing  the  simplest  rule  to 
be  rigorously  true. 

3.  Similar  events  occurred  in  the  history  of  another 
part  of  the  law  of  gravitation  :  namely,  that  the  attrac 
tion  is  proportional  to  the  quantity  of  matter  attracted. 
This  part  of  the  law  may  also  be  thus  stated,  That  the 
weight  of  bodies  arising  from  gravity  is  proportional  to 
their  inertia ;  and  thus,  that  the  accelerating  force  on 
all  bodies  under  the  same  circumstances  is  the  same. 
Newton  made  experiments  which  proved  this  with  re 
gard  to  terrestrial  bodies ;  for  he  found  that,  at  the  end 
of  equal  strings,  balls  of  all  substances,  gold,  silver, 
lead,  glass,  wood,  &c.,  oscillated  in  equal  times'".  But 
a  few  years  ago,  doubts  arose  among  the  German  astro 
nomers  whether  this  law  was  rigorously  true  with  regard 
to  the  planetary  bodies.  Some  calculations  appeared 
to  prove,  that  the  attraction  of  Jupiter  as  shown  by  the 
perturbations  which  he  produces  in  the  small  planets 
Juno,  Vesta,  and  Pallas,  was  different  from  the  attrac 
tion  which  he  exerts  on  his  own  satellites.  Nor  did 
there  appear  to  these  philosophers  anything  inconceiv 
able  in  the  supposition  that  the  attraction  of  a  planet 
might  be  thus  elective.  But  when  Mr.  Airy  obtained 
a  more  exact  determination  of  the  mass  of  Jupiter,  as 

*    Prin.  Lib.  in.,  Prop.  (>. 


indicated  by  his  effect  on  his  satellites,  it  was  found 
that  this  suspicion  was  unfounded  ;  and  that  there  was, 
in  this  case,  no  exception  to  the  universality  of  the  rule, 
that  this  cosmical  attraction  is  in  the  proportion  of  the 
attracted  mass. 

4.  Again :  when    it   had   thus   been  shown   that  a 
mutual  attraction  of  parts,  according  to  the  law  above 
mentioned,  prevailed  throughout  the  extent  of  the  solar 
system,  it  might  still  be  doubted  whether  the  same  law 
extended    to  other  regions  of  the  universe.     It  might 
have  been  perhaps  imagined  that  each  fixed  star  had 
its  peculiar  law  of  force.     But  the  examination  of  the 
motions  of  double  stars  about  each  other,  by  the  two 
Ilerschels  and  others,  appears  to  show  that  these  bodies 
describe  ellipses  as  the  planets  do ;  and  thus  extends  the 
law  of  the  inverse  squares  to  parts  of  the  universe  im 
measurably  distant  from  the  whole  solar  system. 

5.  Since   every  doubt  which  has   been   raised  with 
regard   to  the  universality  and  accuracy  of  the  law  of 
gravitation,  has  thus  ended  in  confirming  the  rule,  it  is 
not  surprizing  that  men's  minds  should  have  returned 
with  additional  force  to  those  views  which  had  at  first 
represented  the  law  as  a  necessary  truth,  capable  of  being 
established  by  reason  alone.     When  it  had  been  proved 
by  Newton  that  gravity  is  really  a  universal  attribute  of 
matter  as  far  as  we  can  learn,  his  pupils  were  not  con 
tent  without  maintaining  it  to  be  an  essential  quality. 
This  is  the  doctrine  held  by  Cotes  in  the  preface  to  the 
second  edition  of  the  Principia  (1712):  "Gravity,"  he 
says,  "  is  a  primary  quality  of  bodies,  as  extension,  mo 
bility,   and  impenetrability  are."     But  Newton  himself 
by  no  means  went  so  far.     In    his   second    Letter   to 
Bentley    (1G03),    he   says:    "You   sometimes  speak  of 
gravity  as  essential  and  inherent  to   matter;    pray  do 
not  ascribe  that  notion  to  me.     The  cause  of  gravity," 

VOL.  i.    w.  P.  S 


he  adds,   "  I  do  not  pretend  to  know,  and  would  take 
more  time  to  consider  of  it." 

Cotes  maintains  his  opinion  by  urging,  that  we  learn 
by  experience  that  all  bodies  possess  gravity,  and  that  we 
do  not  learn  in  any  other  way  that  they  are  extended, 
movcable,  or  solid.  But  we  have  already  seen,  that  the 
ideas  of  space,  time,  and  reaction,  on  which  depend 
extension,  mobility,  and  solidity,  are  not  results,  but 
conditions,  of  experience.  We  cannot  conceive  a  body 
except  as  extended ;  we  cannot  conceive  it  to  exert 
mechanical  action  except  with  some  kind  of  solidity. 
But  so  far  as  our  conceptions  of  body  have  hitherto 
been  developed,  we  find  no  difficulty  in  conceiving  two 
bodies  which  do  not  attract  each  other. 

G.  Newton  lays  down,  in  the  second  edition  of  the 
Princijria,  this  "  Rule  of  Philosophizing"  (Book  in.) ; 
that  "  The  qualities  of  bodies  which  cannot  be  made 
more  or  less  intense,  and  which  belong  to  all  bodies  on 
which  we  are  able  to  make  experiments,  are  to  be  held 
to  be  qualities  of  all  bodies  in  general."  And  this  Rule 
is  cited  in  the  sixth  Proposition  of  the  Third  Book  of 
the  Principia,  (Cor.  2,)  in  order  to  prove  that  gravity, 
proportional  to  the  quantity  of  matter,  may  be  asserted 
to  be  a  quality  of  all  bodies  universally.  But  we  may 
remark  that  a  Rule  of  Philosophizing,  itself  of  precarious 
authority,  cannot  authorize  us  in  ascribing  universality 
to  an  empirical  result.  Geometrical  and  statical  pro 
perties  are  seen  to  be  necessary,  and  therefore  universal : 
but  Newton  appears  disposed  to  assert  a  like  universality 
of  gravity,  quite  unconnected  with  any  necessity.  It 
would  be  a  very  inadequate  statement,  indeed  a  false 
representation,  of  statical  truth,  if  we  were  to  say,  that 
because  every  body  which  has  hitherto  been  tried  has 
been  found  to  have  a  center  of  gravity,  we  venture  to 
assert  that  all  bodies  whatever  have  a  center  of  gravity. 


And  if  we  arc  ever  able  to  assert  the  absolute  univer 
sality  of  the  law  of  gravitation,  we  shall  have  to  rest 
this  truth  upon  the  clearer  developement  of  our  ideas  of 
matter  and  force ;  not  upon  a  Rule  of  Philosophizing, 
which,  till  otherwise  proved,  must  be  a  mere  rule  of 
prudence,  and  which  the  opponent  may  refuse  to  admit. 
7.  Other  persons,  instead  of  asserting  gravity  to  be 
in  its  own  nature  essential  to  matter,  have  made  hypo 
theses  concerning  some  mechanism  or  other,  by  which 
this  mutual  attraction  of  bodies  is  produced"-.  Thus 
the  Cartesians  ascribed  to  a  vortex  the  tendency  of 
bodies  to  a  center ;  Newton  himself  seems  to  have  been 
disposed  to  refer  this  tendency  to  the  elasticity  of  an 
ether;  Le  Sage  propounded  a  curious  hypothesis,  in 
which  this  attraction  is  accounted  for  by  the  impulse 
of  infinite  streams  of  particles  flowing  constantly  through 
the  universe  in  all  directions.  In  these  speculations, 
the  force  of  gravity  is  resolved  into  the  pressure  or  im 
pulse  of  solids  or  fluids.  On  the  other  hand,  hypotheses 
have  been  propounded,  in  which  the  solidity,  and  other 
physical  qualities  of  bodies,  have  been  explained  by 
representing  the  bodies  as  a  collection  of  points,  from 
which  points,  repulsive,  as  well  as  attractive,  forces 
emanate.  This  view  of  the  constitution  of  bodies  was 
maintained  and  developed  by  Boscovich,  and  is  hence 
termed  "  Boscovich's  Theory :"  and  the  discussion  of  it 
will  more  properly  come  under  our  review  at  a  future 
period,  when  we  speak  of  the  question  whether  bodies 
are  made  up  of  atoms.  But  we  may  observe,  that  New 
ton  himself  appears  to  have  inclined,  as  his  followers 
certainly  did,  to  this  mode  of  contemplating  the  physical 
properties  of  bodies.  In  his  Preface  to  the  Principia, 
after  speaking  of  the  central  forces  which  are  exhibited 

*  Sec  Vince,  Observations  on  the  Hyjyothcsis  respecting  Gravitation, 
and  the  Critique  of  that  work,  Edinb.  Rev.  Vol.  xui. 



in  cosmical  phenomena,  he  says :  "  Would  that  we  could 
derive  the  other  phenomena  of  Nature  from  mechanical 
principles  by  the  same  mode  of  reasoning.  For  many 
things  move  me,  so  that  I  suspect  all  these  phenomena 
may  depend  upon  certain  forces,  by  which  the  particles 
of  bodies,  through  causes  not  yet  known,  are  either  im 
pelled  to  each  other  and  cohere  according  to  regular 
figures,  or  are  repelled  and  recede  from  each  other : 
which  forces  being  unknown,  philosophers  have  hitherto 
made  their  attempts  upon  nature  in  vain." 

8.  But  both  these  hypotheses ; — that  by  which  cohe 
sion  and  solidity  are  reduced  to  attractive  and  repulsive 
forces,  and  that  by  which  attraction  is  reduced  to  the 
impulse  and  pressure  of  media; — are  hitherto  merely 
modes  of  representing  mechanical  laws  of  nature ;  and 
cannot,  either  of  them,  be  asserted  as  possessing  any  evi 
dent  truth  or  peremptory  authority  to  the  exclusion  of 
the  other.  This  consideration  may  enable  us  to  estimate 
the  real  weight  of  the  difficulty  felt  in  assenting  to  the 
mutual  attraction  of  bodies  not  in  contact  with  each 
other  ;  for  it  is  often  urged  that  this  attraction  of  bodies 
at  a  distance  is  an  absurd  supposition. 

The  doctrine  is  often  thus  stigmatized,  both  by  popu 
lar  and  by  learned  writers.  It  was  long  received  as  a 
maxim  in  philosophy  (as  Monboddo  informs  us*),  that  a 
body  cannot  act  where  it  is  not,  any  more  than  when  it 
is  not.  But  to  this  we  reply,  that  time  is  a  necessary 
condition  of  our  conception  of  causation,  in  a  different 
manner  from  space.  The  action  of  force  can  only  be 
conceived  as  taking  place  in  a  succession  of  moments,  in 
each  of  which  cause  and  effect  immediately  succeed  each 
other  :  and  thus  the  interval  of  time  between  a  cause  and 
its  remote  effect  is  filled  up  by  a  continuous  succession 
of  events  connected  by  the  same  chain  of  causation.  But 

Ancient  Metaphysics,  Vol.  n.  p.  17"). 


in  space,  there  is  no  such  visible  necessity  of  continuity  ; 
the  action  and  reaction  may  take  place  at  a  distance  from 
each  other ;  all  that  is  necessary  being  that  they  be 
equal  and  opposite. 

Undoubtedly  the  existence  of  attraction  is  rendered 
more  acceptable  to  common  apprehension  by  supposing 
some  intermediate  machinery, — a  cord,  or  rod,  or  fluid, 
— by  which  the  forces  may  be  conveyed  from  one  point 
to  another.  But  such  images  are  rather  fitted  to  satisfy 
those  prejudices  which  arise  from  the  earlier  application 
of  our  ideas  of  force,  than  to  exhibit  the  real  nature  of 
those  ideas.  If  we  suppose  two  bodies  to  pull  each  other 
by  means  of  a  rod  or  a  cord,  we  only  suppose,  in  addition 
to  those  equal  and  opposite  forces  acting  upon  the  two 
bodies  which  forces  are  alone  essential  to  mutual  attrac 
tion,  a  certain  power  of  resisting  transverse  pressure  at 
every  point  of  the  intermediate  line :  which  additional 
supposition  is  entirely  useless,  and  quite  unconnected 
with  the  essential  conditions  of  the  case.  When  the  New 
tonians  were  accused  of  introducing  into  philosophy  an 
unknown  cause  which  they  termed  attraction,  they  justly 
replied  that  they  knew  as  much  respecting  attraction 
as  their  opponents  did  about  impulse.  In  each  case  we 
have  a  knowledge  of  the  conception  in  question  so  far  as 
we  clearly  apprehend  it  under  the  conditions  of  those 
axioms  of  mechanical  causation  which  form  the  basis  of 
our  science  on  such  subjects. 

Having  thus  examined  the  degree  of  certainty  and 
generality  to  which  our  knowledge  of  the  law  of  univer 
sal  gravitation  has  been  carried,  by  the  progress  of 
mechanical  discovery  and  speculation  up  to  the  present 
time,  we  might  proceed  to  the  other  branches  of  science, 
and  examine  in  like  manner  their  grounds  and  conditions. 
But  before  we  do  this,  it  will  be  worth  our  while  to 
attend  for  a  moment  to  the  effect  which  the  progress  of 


mechanical  ideas  among  mathematicians  and  mechanical 
philosophers  has  produced  upon  the  minds  of  other  per 
sons,  who  share  only  in  an  indirect  and  derivative  man 
ner  in  the  influence  of  science. 



1.  WE  have  seen  how  the  progress  of  knowledge 
upon  the  subject  of  motion  and  force  has  produced,  in 
the  course  of  the  world's  history,  a  great  change  in  the 
minds  of  acute  and  speculative  men ;  so  that  such  per 
sons  can  now  reason  with  perfect  steadiness  and  precision 
upon  subjects  on  which,  at  first,  their  thoughts  were 
vague  and  confused ;  and  can  apprehend,  as  truths  of 
complete  certainty  and  evidence,  laws  which  it  required 
great  labour  and  time  to  discover.  This  complete  deve- 
lopement  and  clear  manifestation  of  mechanical  ideas 
has  taken  place  only  among  mathematicians  and  philo 
sophers.  But  yet  a  progress  of  thought  upon  such 
subjects, — an  advance  from  the  obscure  to  the  clear,  and 
from  errour  to  truth, — may  be  traced  in  the  world  at 
large,  and  among  those  who  have  not  directly  cultivated 
the  exact  sciences.  This  diffused  and  collateral  influence 
of  science  manifests  itself,  although  in  a  wavering  and 
fluctuating  manner,  by  various  indications,  at  various 
periods  of  literary  history.  The  opinions  and  reasonings 
which  are  put  forth  upon  mechanical  subjects,  and  above 
all,  the  adoption,  into  common  language,  of  terms  and 
phrases  belonging  to  the  prevalent  mechanical  systems, 
exhibit  to  us  the  most  profound  discoveries  and  specula 
tions  of  philosophers  in  their  effect  upon  more  common 


and  familiar  trains  of  thought.  This  effect  is  by  no 
means  unimportant,  and  we  shall  point  out  some  ex 
amples  of  such  indications  as  we  have  mentioned. 

2.    The  discoveries  of  the   ancients  in  speculative 
mechanics   were,  as   we   have  seen,  very  scanty ;  and 
hardly  extended  their  influence  to  the  unmathematical 
world.     Yet  the  familiar  use  of  the  term  "  center  of 
of  gravity"  preserved  and  suggested  the  most  important 
part  of  what  the  Greeks  had  to  teach.    The  other  phrases 
which   they  employed,    as   momentum,   energy,  virtue, 
force,  and  the  like,  never  had  any  exact  meaning,  even 
among   mathematicians ;    and   therefore   never,    in   the 
ancient   world,   became    the  means  of  suggesting  just 
habits  of  thought.     I  have  pointed  out,  in  the  History 
of  Science,  several  circumstances  which  appear  to  denote 
the  general  confusion  of  ideas  which  prevailed  upon 
mechanical    subjects   during   the   times  of  the  Roman 
empire.     I  have  there  taken  as  one  of  the  examples  of 
this  confusion,  the  fable  narrated  by  Pliny  and  others 
concerning  the  echinei's,  a  small  fish,  which  was  said  to 
stop  a  ship  merely  by  sticking  to  it*.     This  story  was 
adduced  as  betraying  the  absence  of  any  steady  appre 
hension  of  the  equality  of  action  and  reaction  ;  since  the 
fish,  except  it  had  some  immoveable  obstacle  to  hold  by, 
must  be  pulled  forward  by  the  ship,  as  much  as  it  pulled 
the  ship  backward.     If  the  writers  who  speak  of  this 
wonder  had  shown  any  perception  of  the  necessity  of 
a  reaction,  either  produced  by  the  rapid  motion  of  the 
fish's  fins  in  the  water,  or  in  any  other  way,  they  would 
not  be  chargeable  with  this  confusion  of  thought ;  but 
from  their  expressions  it  is,  I  think,  evident  that  they 
saw  no  such  necessity  f.    Their  idea  of  mechanical  action 

*  Hist.  Ind.  Sci.  B.  iv.  c.  i.  sect.  2. 

t  Sec  Prof.  Powell,  On  the  Nature  and  Evidence  of  the  Lan-s  of 
Motion.     Reports  of  the  Ashmolean  Society.  Oxford.  1K37-     Professor 


was  not  sufficiently  distinct  to  enable  them  to  sec  the 
absurdity  of  supposing  an  intense  pressure  with  no 
obstacle  for  it  to  exert  itself  against. 

3.  We  may  trace,  in  more  modern  times  also,  indica 
tions  of  a  general  ignorance  of  mechanical  truths.  Thus 
the  phrase  of  shooting  at  an  object  "point-blank,"  im 
plies  the  belief  that  a  cannon-ball  describes  a  path  of 
which  the  first  portion  is  a  straight  line.  This  error 
was  corrected  by  the  true  mechanical  principles  which 
Galileo  and  his  followers  brought  to  light;  but  these 
principles  made  their  way  to  popular  notice,  principally 
in  consequence  of  their  application  to  the  motions  of  the 
solar  system,  and  to  the  controversies  which  took  place 
respecting  those  motions.  Thus  by  far  the  most  power 
ful  argument  against  the  reception  of  the  Copernican 
system  of  the  universe,  was  that  of  those  who  asked, 
Why  a  stone  dropt  from  a  tower  was  not  left  behind  by 
the  motion  of  the  earth  ?  The  answer  to  this  question, 
now  universally  familiar,  involves  a  reference  to  the  true 
doctrine  of  the  composition  of  motions.  Again;  Kepler's 
persevering  and  strenuous  attempts*  to  frame  a  phy 
sical  theory  of  the  universe  were  frustrated  by  his  igno 
rance  of  the  first  law  of  motion,  which  informs  us  that 
a  body  will  retain  its  velocity  without  any  maintaining 
force.  He  proceeded  upon  the  supposition  that  the  sun's 
force  was  requisite  to  keep  up  the  motion  of  the  planets, 

Powell  has  made  an  objection  to  my  use  of  this  instance  of  confusion 
of  thought ;  the  remark  in  the  text  seems  to  me  to  justify  what  1  said 
in  the  History.  As  an  evidence  that  the  fish  was  not  supposed  to  pro 
duce  its  effect  by  its  muscular  power  acting  on  the  water,  we  may  take 
what  Pliny  says,  Nat.  Hist.,  xxxii.  l,"T)omat  mundi  rabiem,  nullo 
siio  labore ;  non  retinendo,  aut  alio  modo  quam  adluerendo  :"  and  also  he  states  in  another  place  (ix.  41,)  that  when  it  is  preserved  in 
pickle,  it  may  be  used  in  recovering  gold  which  has  fallen  into  a  deep 
well.  All  tins  implies  adhesion  alone,  with  no  conception  of  reaction. 
*  11'*!.  I,,'!,  ,SW.,  B.  .,.  c.  iv..  ;,n,l  B.  vn.  c.  i. 

DIFFUSION    OF    CLEAR    MECHANICAL    IDEAS.         2()f> 

as  well  as  to  deflect  and  modify  it ;  and  he  was  thus 
led  to  a  system  which  represented  the  sun  as  carrying 
round  the  planets  in  their  orbits  by  means  of  a  vortex, 
produced  by  his  revolution.  The  same  neglect  of  the 
laws  of  motion  presided  in  the  formation  of  Descartes' 
system  of  vortices.  Although  Descartes  had  enunciated 
in  words  the  laws  of  motion,  he  and  his  followers  showed 
that  they  had  not  the  practical  habit  of  referring  to 
these  mechanical  principles ;  and  dared  not  trust  the 
planets  to  move  in  free  space  without  some  surrounding 
machinery  to  support  them*. 

4.  When  at  last  mathematicians,  following  Newton, 
had  ventured  to  consider  the  motion  of  each  planet  as  a 
mechanical  problem  not  different  in  its  nature  from  the 
motion  of  a  stone  cast  from  the  hand ;  and  when  the 
solution  of  this  problem  and  its  immense  consequences 
had  become  matters  of  general  notoriety  and  interest ; 
the  new  views  introduced,  as  is  usual,  new  terms,  which 
soon  became  extensively  current.  We  meet  with  such 
phrases  as  "  flying  off  in  the  tangent,"  and  "  deflexion 
from  the  tangent;"  with  antitheses  between  "centripetal" 
and  "centrifugal  force,"  or  between  "projectile"  and 
"  central  force."  "  Centers  of  force,"  "  disturbing  forces," 
"perturbations,"  and  "perturbations  of  higher  orders," 
arc  not  unfrequently  spoken  of:  and  the  expression  "to 
gravitate,"  and  the  term  "universal  gravitation,"  acquired 
a  permanent  place  in  the  language. 

Yet  for  a  long  time,  and  even  up  to  the  present  day, 
we  find  many  indications  that  false  and  confused  appre 
hensions  on  such  subjects  arc  by  no  means  extirpated. 

*  I  have,  in  the  History,  applied  to  Descartes  the  character  which 
Iiacon  gives  to  Aristotle,  "  Audax  siimil  et  paviclus  :"  though  he  was 
hold  enough  to  enunciate  the  laws  of  motion  without  knowing  them 
aright,  he  had  not  the  courage  to  leave  the  planets  to  describe  their 
orbits  by  the  agency  of  those  laws,  without  the  machinery  of  contact. 


Arguments  arc  urged  against  the  mechanical  system  of 
the  universe,  implying  in  the  opponents  an  absence  of 
all  clear  mechanical  notions.  Many  of  this  class  of 
writers  retrograde  to  Kepler's  point  of  view.  This  is, 
for  example,  the  case  with  Lord  Monboddo,  who,  arguing 
on  the  assumption  that  force  is  requisite  to  maintain,  as 
well  as  to  deflect  motion,  produced  a  series  of  attacks 
upon  the  Newtonian  philosophy ;  which  he  inserted  in 
his  Ancient  Metaphysics,  published  in  1771)  and  the 
succeeding  years.  This  writer  (like  Kepler),  measures 
force  by  the  velocity  which  the  body  has*.,  not  by  that 
which  its  gains.  Such  a  use  of  language  would  prevent 
our  obtaining  any  laws  of  motion  at  all.  Accordingly, 
the  author,  in  the  very  next  page  to  that  which  I  have 
just  quoted,  abandons  this  measure  of  force,  and,  in  cur 
vilinear  motion,  measures  force  by  "  the  fall  from  the 
extremity  of  the  arc."  Again ;  in  his  objections  to  the 
received  theory,  he  denies  that  curvilinear  motion  is 
compounded,  although  his  own  mode  of  considering  such 
motion  assumes  this  composition  in  the  only  way  in 
which  it  was  ever  intended  by  mathematicians.  Many 
more  instances  might  be  adduced  to  show  that  a  want 
of  cultivation  of  the  mechanical  ideas  rendered  this  phi- 
,  losopher  incapable  of  judging  of  a  mechanical  system. 

The  following  extract  from  the  Ancient  Metaphy 
sics,  may  be  sufficient  to  show  the  value  of  the  author's 
criticism  on  the  subjects  of  which  we  are  now  speaking. 
His  object  is  to  prove  that  there  do  not  exist  a  centri 
petal  and  a  centrifugal  force  in  the  case  of  elliptical 
motion.  "Let  any  man  move  in  a  circular  or  elliptical 
line  described  to  him ;  and  he  will  find  no  tendency  in 
himself  cither  to  the  center  or  from  it,  much  less  both. 
If  indeed  he  attempt  to  make  the  motion  with  great 
velocity,  or  if  he  do  it  carelessly  and  inattentively,  he 

*   Anc.  Met.  Vol.  IT.  K  v.  p.  vi..  p.  413. 


may  go  out  of  the  line,  either  towards  the  center  or  from 
it :  hut  this  is  to  be  ascribed,  not  to  the  nature  of  the 
motion,  but  to  our  infirmity ;  or  perhaps  to  the  animal 
form,  which  is  more  fitted  for  progressive  motion  in  a 
right  line  than  for  any  kind  of  curvilinear  motion.  But 
this  is  not  .the  case  with  a  sphere  or  spheroid,  which  is 
equally  adapted  to  motion  in  all  directions'"."  We  need 
hardly  remind  the  reader  that  the  manner  in  which  a 
man  running  round  a  small  circle,  finds  it  necessary  to 
lean  inwards,  in  order  that  there  may  be  a  centripetal 
inclination  to  counteract  the  centrifugal  force,  is  a 
standard  example  of  our  mechanical  doctrines ;  and  this 
fact  (quite  familiar  in  practice  as  well  as  theory,)  is  in 
direct  contradiction  of  Lord  Monboddo's  assertion. 

5.  A  similar  absence  of  distinct  mechanical  thought 
appears  in  some  of  the  most  celebrated  metaphysicians 
of  Germany.  I  have  elsewhere  noted f  the  opinion  ex 
pressed  by  Hegel,  that  the  glory  which  belongs  to  Kepler 
has  been  unjustly  transferred  to  Newton ;  and  I  have 
suggested,  as  the  explanation  of  this  mode  of  thinking, 
that  Hegel  himself,  in  the  knowledge  of  mechanical 
truth,  had  not  advanced  beyond  Kepler's  point  of  view. 
Persons  who  possess  conceptions  of  space  and  number, 
but  who  have  not  learnt  to  deal  with  ideas  of  force  and 
causation,  may  see  more  value  in  the  discoveries  of  Kepler 
than  in  those  of  Newton.  Another  exemplification  of 
this  state  of  mind  may  be  found  in  Mr.  Schelling's  spe 
culations  ;  for  instance,  in  his  Lectures  on  the  Method  of 
Academical  Study.  In  the  twelfth  Lecture,  on  the  Study 
of  Physics  and  Chemistry,  he  says,  (p.  26G,)  "  What  the 
mathematical  natural  philosophy  has  done  for  the  know 
ledge  of  the  laws  of  the  universe  since  the  time  that 
they  were  discovered  by  his  (Kepler's)  godlike  genius,  is, 

*  Anc.  Met.,  Vol.  i.  R  ii.  c.  10,  p.  2(51. 
t    Hisl.  Ind.  Sci..  B.  vn.  c.  ii.  sect.  />. 


as  is  well  known,  this:  it  has  attempted  a  construction 
of  those  laws  which,  according  to  its  foundations,  is  alto 
gether  empirical.  We  may  assume  it  as  a  general  rule, 
that  in  any  proposed  construction,  that  which  is  not  a 
pure  general  form  cannot  have  any  scientific  import 
or  truth.  The  foundation  from  which  the  centrifugal 
motion  of  the  bodies  of  the  world  is  derived,  is  no  ne 
cessary  form,  it  is  an  empirical  fact.  The  Newtonian 
attractive  force,  even  if  it  be  a  necessary  assumption  for 
a  merely  reflective  view  of  the  subject,  is  still  of  no 
significance  for  the  Reason,  which  recognizes  only  abso 
lute  relations.  The  grounds  of  the  Keplerian  laws  can 
be  derived,  without  any  empirical  appendage,  purely 
from  the  doctrine  of  Ideas,  and  of  the  two  Unities,  which 
are  in  themseves  one  Unity,  and  in  virtue  of  which  each 
being,  while  it  is  absolute  in  itself,  is  at  the  same  time 
in  the  absolute,  and  reciprocally." 

It  will  be  observed,  that  in  this  passage  our  mecha 
nical  laws  are  objected  to  because  they  are  not  necessary 
results  of  our  ideas ;  which,  however,  as  we  have  seen, 
according  to  the  opinion  of  some  eminent  mechanical 
philosophers,  they  are.  But  to  assume  this  evident 
necessity  as  a  condition  of  every  advance  in  science,  is 
to  mistake  the  last,  perhaps  unattainable  step,  for  the 
first,  which  lies  before  our  feet.  And,  without  inquiring 
further  about  "  the  Doctrine  of  the  two  Unities,"  or  the 
manner  in  which  from  that  doctrine  we  may  deduce  the 
Keplerian  laws,  we  may  be  well  convinced  that  such  a 
doctrine  cannot  supply  any  sufficient  reason  to  induce  us 
to  quit  the  inductive  path  by  which  all  scientific  truth 
up  to  the  present  time  has  been  acquired. 

G.  But  without  going  to  schools  of  philosophy  oppo 
sed  to  the  Inductive  School,  we  may  find  many  loose  and 
vague  habits  of  thinking  on  mechanical  subjects  among 
the  common  classes  of  readers  and  reasoners.  And 

DIFFUSION    OF    CLEAR    MECHANICAL    IDEAS.         20*9 

there  are  some  familiar  modes  of  employing  the  phrase 
ology  of  mechanical  science,  which  are,  in  a  certain 
degree,  chargeable  with  inaccuracy,  and  may  produce 
or  perpetuate  confusion.  Among  such  cases  we  may 
mention  the  way  in  which  the  centripetal  and  centri 
fugal  forces,  and  also  the  projectile  and  central  forces 
of  the  planets,  are  often  compared  or  opposed.  Such 
antitheses  sometimes  proceed  upon  the  false  notion  that 
the  two  members  of  these  pairs  of  forces  are  of  the 
same  kind :  whereas  on  the  contrary  the  projectile  force 
is  a  hypothetical  impulsive  force  which  may,  at  some 
former  period,  have  caused  the  motion  to  begin ;  while 
the  central  force  is  an  actual  force,  which  must  act  con 
tinuously  and  during  the  whole  time  of  the  motion,  in 
order  that  the  motion  may  go  on  in  the  curve.  In  the 
same  manner  the  centrifugal  force  is  not  a  distinct  force 
in  a  strict  sense,  but  only  a  certain  result  of  the  first 
law  of  motion,  measured  by  the  portion  of  centripetal 
force  wrhich  counteracts  it.  Comparisons  of  quantities 
so  heterogeneous  imply  confusion  of  thought,  and  often 
suggest  baseless  speculations  and  imagined  reforms  of 
the  received  opinions. 

7.  I  might  point  out  other 'terms  and  maxims,  in 
addition  to  those  already  mentioned,  which,  though  for 
merly  employed  in  a  loose  and  vague  manner,  are  now 
accurately  understood  and  employed  by  all  just  thinkers; 
and  thus  secure  and  diffuse  a  right  understanding  of  me 
chanical  truths.  Such  are  momentum,  inertia,  quantity 
of  matter,  quantity  of  motion  ;  thai  force  is  proportional 
to  its  effects;  that  action  and  reaction  are  equal;  that 
what  is  gained  in  force  by  machinery  is  lost  in  time; 
that  the  quantity  of  motion  in  the  world  cannot  be  either 
increased  or  diminished.  When  the  expression  of  the 
truth  thus  becomes  easy  and  simple,  clear  and  con 
vincing,  the  meanings  given  to  words  and  phrases  by 


discoverers  glide  into  the  habitual  texture  of  men's  rea 
sonings,  and  the  effect  of  the  establishment  of  true 
mechanical  principles  is  felt  far  from  the  school  of  the 
mechanician.  If  these  terms  and  maxims  are  understood 
with  tolerable  clearness,  they  carry  the  influence  of 
truth  to  those  who  have  no  direct  access  to  its  sources. 
Many  an  extravagant  project  in  practical  machinery,  and 
many  a  wild  hypothesis  in  speculative  physics,  has  been 
repressed  by  the  general  currency  of  such  maxims  as  we 
have  just  quoted. 

8.  Indeed  so  familiar  and  evident  are  the  elementary 
truths  of  mechanics  when  expressed  in  this  simple  form, 
that  they  are  received  as  truisms ;  and  men  are  disposed 
to  look  back  with  surprize  and  scorn  at  the  speculations 
which  were  carried  on  in  neglect  of  them.  The  most 
superficial  reasoner  of  modern  times  thinks  himself  enti 
tled  to  speak  with  contempt  and  ridicule  of  Kepler's 
hypothesis  concerning  the  physical  causes  of  the  celestial 
motions:  and  gives  himself  credit  for  intellectual  supe 
riority,  because  he  sees,  as  self-evident,  what  such  a  man 
could  not  discover  at  all.  It  is  well  for  such  a  person  to 
recollect,  that  the  real  cause  of  his  superior  insight  is 
not  the  pre-eminence  of  his  faculties,  but  the  successful 
labours  of  those  who  have  preceded  him.  The  language 
which  he  has  learnt  to  use  unconsciously,  has  been 
adapted  to,  and  moulded  on,  ascertained  truths.  When 
he  talks  familiarly  of  "  accelerating  forces"  and  "  de 
flexions  from  the  tangent,"  he  is  assuming  that  which 
Kepler  did  not  know,  and  which  it  cost  Galileo  and  his 
disciples  so  much  labour  and  thought  to  establish.  Lan 
guage  is  often  called  an  instrument  of  thought ;  but  it 
is  also  the  nutriment  of  thought ;  or  rather,  it  is  the 
atmosphere  in  which  thought  lives :  a  medium  essential 
to  the  activity  of  our  speculative  power,  although  invi 
sible  and  imperceptible  in  its  operation ;  and  an  element 


modifying,  by  its  qualities  and  changes,  the  growth  and 
complexion  of  the  faculties  which  it  feeds.  In  this  way 
the  influence  of  preceding  discoveries  upon  subsequent 
ones,  of  the  past  upon  the  present,  is  most  penetrating 
and  universal,  though  most  subtle  and  difficult  to  trace. 
The  most  familiar  words  and  phrases  are  connected  by 
imperceptible  ties  with  the  reasonings  and  discoveries  of 
former  men  and  distant  times.  Their  knowledge  is  an 
inseparable  part  of  ours ;  the  present  generation  inherits 
and  uses  the  scientific  wealth  of  all  the  past.  And  this 
is  the  fortune,  not  only  of  the  great  and  rich  in  the 
intellectual  world :  of  those  who  have  the  key  to  the 
ancient  storehouses,  and  who  have  accumulated  treasures 
of  their  own; — but  the  humblest  inquirer,  while  he 
puts  his  reasonings  into  words,  benefits  by  the  labours 
of  the  greatest  discoverers.  When  he  counts  his  little 
wealth,  he  finds  that  he  has  in  his  hands  coins  which 
bear  the  image  and  superscription  of  ancient  and  modern 
intellectual  dynasties ;  and  that  in  virtue  of  this  posses 
sion,  acquisitions  are  in  his  power,  solid  knowledge 
within  his  reach,  which  none  could  ever  have  attained 
to,  if  it  were  not  that  the  gold  of  truth,  once  dug  out  of 
the  mine,  circulates  more  and  more  widely  among  man 

9.  Having  so  fully  examined,  in  the  preceding  in 
stances,  the  nature  of  the  progress  of  thought  which 
science  implies,  both  among  the  peculiar  cultivators  of 
science,  and  in  that  wider  world  of  general  culture  which 
receives  only  an  indirect  influence  from  scientific  disco 
veries,  we  shall  not  find  it  necessary  to  go  into  the  same 
extent  of  detail  with  regard  to  the  other  provinces  of 
human  knowledge.  In  the  case  of  the  Mechanical 
Sciences,  we  have  endeavoured  to  show,  not  only  that 
Ideas  are  requisite  in  order  to  form  into  a  science  the 
Facts  which  nature  otters  to  us,  but  that  we  can  advance, 


almost  or  quite,  to  a  complete  identification  of  the  Facts 
with  the  Ideas.  In  the  sciences  to  which  we  now  pro 
ceed,  we  shall  not  seek  to  fill  up  the  chasm  by  which 
Facts  and  Ideas  are  separated ;  but  we  shall  endeavour 
to  detect  the  Ideas  which  our  knowledge  involves,  to 
show  how  essential  these  arc ;  and  in  some  respects  to 
trace  the  mode  in  which  they  have  been  gradually  de 
veloped  among  men. 

10.  The  motions  of  the  heavenly  bodies,  their  laws, 
their  causes,  are  among  the  subjects  of  the  first  division 
of  the  Mechanical  Sciences ;  and  of  these  sciences  we 
formerly  sketched  the  history,  and  have  now  endeavoured 
to  exhibit  the  philosophy.  If  we  were  to  take  any  other 
class  of  motions,  their  laws  and  causes  might  give  rise 
to  sciences  which  would  be  mechanical  sciences  in  exactlv 


the  same  sense  in  which  Physical  Astronomy  is  so.  The 
phenomena  of  magnets,  of  electrical  bodies,  of  galva- 
nical  apparatus,  seem  to  form  obvious  materials  for  such 
sciences ;  and  if  they  were  so  treated,  the  philosophy  of 
such  branches  of  knowledge  would  naturally  come  under 
our  consideration  at  this  point  of  our  progress. 

But  on  looking  more  attentively  at  the  sciences  of 
Electricity,  Magnetism,  and  Galvanism,  we  discover 
cogent  reasons  for  transferring  them  to  another  part  of 
our  arrangement ;  we  find  it  advisable  to  associate  them 
with  Chemistry,  and  to  discuss  their  principles  when 
we  can  connect  them  with  the  principles  of  chemical 
science.  For  though  the  first  steps  and  narrower  gene 
ralizations  of  these  sciences  depend  upon  mechanical 
ideas,  the  highest  laws  and  widest  generalizations  which 
we  can  reach  respecting  them,  involve  chemical  rela 
tions.  The  progress  of  these  portions  of  knowledge  is 
in  some  respects  opposite  to  the  progress  of  Physical 
Astronomy.  In  this,  we  begin  with  phenomena  which 
appear  to  indicate  peculiar  and  various  qualities  in  the 


bodies  which  we  consider,  (namely,  the  heavenly  bodies,) 
and  we  find  in  the  end  that  all  these  qualities  resolve 
themselves  into  one  common  mechanical  property,  which 
exists  alike  in  all  bodies  and  parts  of  bodies.  On  the 
contrary,  in  studying  magnetical  and  electrical  laws,  we 
appear  at  first  to  have  a  single  extensive  phenomenon, 
attraction  and  repulsion :  but  in  our  attempts  to  gene 
ralize  this  phenomenon,  we  find  that  it  is  governed  by 
conditions  depending  upon  something  quite  separate 
from  the  bodies  themselves,  upon  the  presence  and  dis 
tribution  of  peculiar  and  transitory  agencies ;  and,  so  far 
as  we  can  discover,  the  general  lawrs  of  these  agencies 
are  of  a  chemical  nature,  and  are  brought  into  action  by 
peculiar  properties  of  special  substances.  In  cosmical 
phenomena,  everything,  in  proportion  as  it  is  referred  to 
mechanical  principles,  tends  to  simplicity, — to  permanent 
uniform  forces, — to  one  common,  positive,  property.  In 
magnetical  and  electrical  appearances,  on  the  contrary, 
the  application  of  mechanical  principles  leads  only  to 
a  new  complexity,  which  requires  a  new  explanation; 
and  this  explanation  involves  changeable  and  various 
forces, — gradations  and  oppositions  of  qualities.  The 
doctrine  of  the  universal  gravitation  of  matter  is  a  simple 
and  ultimate  truth,  in  which  the  mind  can  acquiesce 
and  repose.  We  rank  gravity  among  the  mechanical 
attributes  of  matter,  and  we  see  no  necessity  to  derive 
it  from  any  ulterior  properties.  Gravity  belongs  to  mat 
ter,  independent  of  any  conditions.  But  the  conditions  of 
magnetic  or  electrical  activity  require  investigation  as 
much  as  the  lares  of  their  action.  Of  these  conditions 
no  mere  mechanical  explanation  can  be  given ;  we  are 
compelled  to  take  along  with  us  chemical  properties 
and  relations  also :  and  thus  magnetism,  electricity,  gal 
vanism,  are  mechanico-chemical  sciences. 

11.    Before  considering  these,  therefore,  I  shall  treat 

VOL.  I.     W.  P.  T 


of  what  I  shall  call  Secondary  Mechanical  Sciences ;  by 
which  expression  I  mean  the  sciences  depending  upon 
certain  qualities  which  our  senses  discover  to  us  in 
bodies; — Optics,  which  has  visible  phenomena  for  its 
subject;  Acoustics,  the  science  of  hearing;  the  doctrine 
of  Ih'ftt,  a  quality  which  our  touch  recognizes :  to  this 
last  science  1  shall  take  the  liberty  of  sometimes  giving 
the  name  T/iermotics,  analogous  to  the  names  of  the 
other  two.  If  our  knowledge  of  the  phenomena  of  Smell 
and  Taste  had  been  successfully  cultivated  and  syste 
matized,  the  present  part  of  our  work  would  be  the 
place  for  the  philosophical  discussion  of  those  sensations 
as  the  subjects  of  science. 

The  branches  of  knowledge  thus  grouped  in  one  class 
involve  common  Fundamental  Ideas,  from  which  their 
principles  are  derived  in  a  mode  analogous,  at  least  in 
a  certain  degree,  to  the  mode  in  which  the  principles  of 
the  mechanical  sciences  are  derived  from  the  funda 
mental  ideas  of  causation  and  reaction.  We  proceed 
now  to  consider  these  Fundamental  Ideas,  their  nature, 
development,  and  consequences. 


THE  Axiom  that  Reaction  is  equal  and  opposite  to  Action^  may  appear 
to  be  at  variance  with  a  maxim  concerning  Cause  which  is  commonly 
current ;  namely,  that  the  "  Cause  precedes  Effect,  and  Effect  follows 
Cause."  For  it  may  he  said,  if  J.,  the  Action,  and  -R,  the  Reaction,  can 
be  considered  as  mutually  the  cause  of  each  other,  A  must  precede  R, 
and  yet  must  follow  it,  which  is  impossible.  But  to  this  I  reply,  that 
in  those  cases  of  direct  Causation  to  which  the  maxim  applies,  the  Cause 
and  Effect  are  not  successive,  but  simultaneous.  If  I  press  against  some 
obstacle,  the  obstacle  resists  and  returns  the  pressure  at  the  instant  it  is 
exerted,  not  after  any  interval  of  time,  however  small.  The  common 

NOTES    ON    CHAPTERS    IV.    AND    VI.  275 

maxim,  that  the  effect  follows  the  cause,  has  arisen  from  the  practice  of 
considering,  as  examples  of  cause  and  effect,  not  instantaneous  forces  or 
causes,  and  the  instantaneous  changes  which  they  produce ;  hut  taking, 
instead  of  this  latter,  the  cumulative  effects  produced  in  the  course  of 
time,  and  compared  with  like  results  occurring  without  the  action  of  the 
cause.  Thus,  if  we  alter  the  length  of  a  clock-pendulum,  this  change 
produces,  as  its  effect,  a  subsequent  change  of  rate  in  the  clock  :  because 
the  rate  is  measured  by  the  accumulated  effects  of  the  pendulum's  gravity, 
before  and  after  the  change.  But  the  pendulum  produces  its  mechanical 
effect  upon  the  escapement,  at  the  moment  of  its  contact,  and  each 
wheel  upon  the  next,  at  the  moment  of  its  contact.  As  has  been  said 
in  a  Review  of  this  work,  "  The  time  lost  in  cases  of  indirect  physical 
causation  is  consumed  in  the  movements  which  take  place  among  the 
parts  of  the  mechanism  in  action,  by  which  the  active  forces  so  trans 
formed  into  momentum  are  transported  over  intervals  of  space  to  new 
points  of  action,  the  motion  of  matter  in  such  cases  being  regarded  as  a 
mere  carrier  of  force."  (Quarterly  Rev.,  No.  cxxxv.,  p.  212.)  See  this 
subject  further  treated  in  a  Memoir  entitled,  "  Discussion  of  the  Ques 
tion  : — Are  Cause  and  Effect  Successive  or  Simultaneous  ?"  in  the 
Memoirs  of  the  Cambridge  Philosophical  Society,  Vol.  vii.  Part  Hi. 


To  the  doctrine  that  mechanical  principles,  such  as  the  one  here  under 
consideration  (that  the  pressure  on  the  point  of  support  is  equal  to  the 
sum  of  the  weights),  are  derived  from  our  Ideas,  and  do  not  flow  from 
but  regulate  our  experience,  objections  are  naturally  made  by  those  who 
assert  all  our  knowledge  to  be  derived  from  experience.  How,  they  ask, 
can.  we  know  the  properties  of  pressures,  levers  and  the  like,  except 
from  experience?  What  but  experience  can  possibly  inform  us  that  a 
force  applied  transversely  to  a  lever  will  have  any  tendency  to  tuni  the 
lever  on  its  center  ?  This  cannot  be,  except  we  suppose  in  the  lever 
tenacity,  rigidity  and  the  like,  which  are  qualities  known  only  by 
experience.  And  it  is  obvious  that  this  line  of  argument  might  be 
carried  on  through  the  whole  subject. 

My  answer  to  this  objection  is  a  remark  of  the  same  kind  as  one 
which  I  have  made  respecting  the  Ideas  of  Space,  Time,  and  Number, 
in  a  Note  at  the  end  of  Chapter  x.  of  the  last  Book.  The  mind,  in 
apprehending  events  as  causes  and  effects,  is  govenied  by  Laws  of  its 
own  Activity  ;  and  these  Laws  govern  the  results  of  the  mind's  action  ; 



and  make  these  results  conform  to  the  Axioms  of  Causation.  But  this 
activity  of  the  mind  is  awakened  and  developed  by  being  exercised ; 
and  in  dealing  with  the  examples  of  cause  and  effect  here  spoken  of, 
(namely,  pressure  and  resistance,  force  and  motion,)  the  mind's  activity 
is  necessarily  governed  also  by  the  bodily  powers  of  perception  and 
action.  We  are  human  beings  only  in  so  far  as  we  have  existed  in  space 
and  time,  and  of  our  human  faculties,  developed  by  our  existence  in  space 
and  time,  space  and  time  are  necessary  conditions.  In  like  manner,  we 
are  human  beings  only  in  so  far  as  we  have  bodies,  and  bodily  organs ; 
and  our  bodies  necessarily  imply  material  objects  external  to  us.  And 
hence  our  human  faculties,  developed  by  our  bodily  existence  in  a 
material  world,  have  the  conditions  of  matter  for  their  necessary  Laws. 
I  have  already  said  (Chap,  v.)  that  our  conception  of  Force  arises  with 
our  consciousness  of  our  own  muscular  exertions ; — that  Force  cannot 
be  conceived  without  Resistance  to  exercise  itself  upon ; — and  that  this 
resistance  is  supplied  by  Matter.  And  thus  the  conception  of  Matter, 
and  of  the  most  general  modes  in  which  Matter  receives,  resists,  and 
transmits  force,  are  parts  of  our  constitution  which,  though  awakened 
and  unfolded  by  our  being  in  a  material  world,  are  not  distinguishable 
from  the  original  structure  of  the  mind.  I  do  not  ascribe  to  the 
mind  Ideas  which  it  would  have,  even  if  it  had  no  intercourse  with 
the  world  of  space,  time,  and  matter;  because  we  cannot  imagine  a 
mind  in  such  a  state.  But  I  attempt  to  point  out  and  classify  those 
Conditions  of  all  Experience,  to  which  the  intercourse  of  all  minds  with 
the  material  world  has  necessarily  given  rise  in  all.  Truths  thus  neces 
sarily  acquired  in  the  course  of  all  experience,  cannot  be  said  to  be 
learnt  from  experience,  in  the  same  sense  in  which  particular  facts,  at 
definite  times,  are  learnt  from  experience,  learnt  by  some  persons  and 
not  by  others,  learnt  with  more  or  less  of  certainty.  These  latter 
special  truths  of  experience  will  be  very  important  subjects  of  our  con 
sideration;  but  our  whole  chance  of  discussing  them  with  any  profit 
depends  upon  our  keeping  them  distinct  from  the  necessary  and  uni 
versal  conditions  of  experience.  Here,  as  everywhere,  we  must  keep 
in  view  the  fundamental  antithesis  of  Ideas  and  Facts. 






1 .  Of  Primary  and  Secondary  Qualities. — IN  the 
same  way  in  which  the  mechanical  sciences  depend  upon 
the  Idea  of  Cause,  and  have  their  principles  regulated 
by  the  development  of  that  Idea,  it  will  be  found  that 
the  sciences  which  have  for  their  subject  Sound,  Light, 
and  Heat,  depend  for  their  principles  upon  the  Funda 
mental  Idea  of  Media  by  means  of  which  we  perceive 
those  qualities.  Like  the  idea  of  cause,  this  idea  of  a 
medium  is  unavoidably  employed,  more  or  less  distinctly, 
in  the  common,  unscientific  operations  of  the  under 
standing;  and  is  recognized  as  an  express  principle  in 
the  earliest  speculative  essays  of  man.  But  here  also, 
as  in  the  case  of  the  mechanical  sciences,  the  develope- 
ment  of  the  idea,  and  the  establishment  of  the  scientific 
truths  which  depend  upon  it,  was  the  business  of  a 
succeeding  period,  and  was  only  executed  by  means  of 
long  and  laborious  researches,  conducted  with  a  constant 
reference  to  experiment  and  observation. 

Among  the  most  prominent  manifestations  of  the 
influence  of  the  idea  of  a  medium  of  which  we  have 
now  to  speak,  is  the  distinction  of  the  qualities  into 


primary,  and  secondary  qualities.  This  distinction  has 
been  constantly  spoken  of  in  modern  times :  yet  it  has 
often  been  a  subject  of  discussion  among  metaphysicians 
whether  there  be  really  such  a  distinction,  and  what  the 
true  difference  is.  Locke  states  it  thus* :  original  or 
primary  qualities  of  bodies  are  "  such  as  are  utterly  in 
separable  from  the  body  in  what  estate  soever  it  may 
be, — such  as  sense  constantly  finds  in  every  particle  of 
matter  which  has  bulk  enough  to  be  perceived,  and  the 
mind  finds  inseparable  from  every  particle  of  matter, 
though  less  than  to  make  itself  singly  perceived  by  our 
senses :"  and  he  enumerates  them  as  solidity,  extension, 
figure,  motion  or  rest,  and  number.  Secondary  qualities, 
on  the  other  hand,  are  such  "which  in  truth  are  nothing 
in  the  objects  themselves,  but  powers  to  produce  various 
sensations  in  us  by  their  primary  qualities,  i.  e.,  by  the 
bulk,  figure,  texture,  and  motion  of  their  insensible 
parts,  as  colours,  sounds,  tastes,  &c." 

Dr.  Reidf,  reconsidering  this  subject,  puts  the  differ 
ence  in  another  way.  There  is,  he  says,  a  real  foundation 
for  the  distinction  of  primary  and  secondary  qualities, 
and  it  is  this :  "  That  our  senses  give  us  a  direct  and  dis 
tinct  notion  of  the  primary  qualities,  and  inform  us  what 
they  are  in  themselves ;  but  of  the  secondary  qualities, 
our  senses  give  us  only  a  relative  and  obscure  notion. 
They  inform  us  only  that  they  are  qualities  that  affect  us 
in  a  certain  manner,  that  is,  produce  in  us  a  certain  sen 
sation  ;  but  as  to  what  they  are  in  themselves,  our  senses 
leave  us  in  the  dark." 

Dr.  Brown  |  states  the  distinction  somewhat  other 
wise.  We  give  the  name  of  matter,  he  observes,  to  that 
which  has  extension  and  resistance :  these,  therefore,  are 
primary  qualities  of  matter,  because  they  compose  our 

*  Essa//,  B.  n.  ch.  viii,  s.  9,  10,  t    Essays,  B.  n.  c.  xvii. 

.|   Lectures,  n.  12. 

OF   THE    IDEA    OF   A    MEDIUM.  279 

definition  of  it.  All  other  qualities  are  secondary,  since 
they  are  ascribed  to  bodies  only  because  we  find  them 
associated  with  the  primary  qualities  which  form  our 
notion  of  those  bodies. 

It  is  not  necessary  to  criticize  very  strictly  these  vari 
ous  distinctions.  If  it  were,  it  would  be  easy  to  find 
objections  to  them.  Thus  Locke,  it  may  be  observed, 
does  not  point  out  any  reason  for  believing  that  his 
secondary  qualities  are  produced  by  the  primary.  How 
are  we  to  learn  that  the  colour  of  a  rose  arises  from  the 
bulk,  figure,  texture,  and  motion  of  its  particles  ?  Cer 
tainly  our  senses  do  not  teach  us  this;  and  in  what  other 
way,  on  Locke's  principles,  can  we  learn  it?  Reid's 
statement  is  not  more  free  from  the  same  objection. 
How  does  it  appear  that  our  notion  of  Warmth  is  rela 
tive  to  our  own  sensations  more  than  our  notion  of 
Solidity  ?  And  if  we  take  Brown's  account,  we  may  still 
ask  whether  our  selection  of  certain  qualities  to  form 
our  idea  and  definition  of  matter  be  arbitrary  and  with 
out  reason?  If  it  be,  how  can  it  make  a  real  distinction? 
if  it  be  not,  what  is  the  reason  ? 

I  do  not  press  these  objections,  because  I  believe  that 
any  of  the  above  accounts  of  the  distinction  of  primary 
and  secondary  qualities  is  right  in  the  main,  however 
imperfect  it  may  be.  The  difference  between  such 
qualities  as  Extension  and  Solidity  on  the  one  hand, 
and  Colour  or  Fragrance  on  the  other,  is  assented  to 
by  all,  with  a  conviction  so  firm  and  indestructible,  that 
there  must  be  some  fundamental  principle  at  the  bottom 
of  the  belief,  however  difficult  it  may  be  to  clothe  the 
principle  in  words.  That  successive  efforts  to  express 
the  real  nature  of  the  difference  were  made  by  men  so 
clear-sighted  and  acute  as  those  whom  I  have  quoted, 
even  if  none  of  them  are  satisfactory,  shows  how  strong 
and  how  deeply-seated  is  the  perception  of  truth  which 
impels  us  to  such  attempts. 


The  most  obvious  mode  of  stating  the  difference  of 
primary  and  secondary  qualities,  as  it  naturally  offers 
itself  to  speculative  minds,  appears  to  be  that  employed 
bv  Locke,  slightly  modified.  Certain  of  the  qualities  of 
bodies,  as  their  bulk,  figure,  and  motion,  are  perceived 
immediately  in  the  bodies  themselves.  Certain  other 
qualities  as  sound,  colour,  heat,  are  perceived  by  means 
of  some  medium.  Our  conviction  that  this  is  the  case 
is  spontaneous  and  irresistible ;  and  this  difference  of 
qualities  immediately  and  mediately  perceived  is  the  dis 
tinction  of  primary  and  secondary  qualities.  We  proceed 
further  to  examine  this  conviction. 

2.  The  Idea  of  Externality. — In  reasoning  concern 
ing  the  secondary  qualities  of  bodies,  we  are  led  to  assume 
the  bodies  to  be  external  to  us,  and  to  be  perceived  by 
means  of  some  medium  intermediate  between  us  and 
them.  These  assumptions  are  fundamental  conditions 
of  perception,  inseparable  from  it  even  in  thought. 

That  objects  are  external  to  us,  that  they  are  without 
us,  that  they  have  outness,  is  as  clear  as  it  is  that  these 
words  have  any  meaning  at  all.  This  conviction  is,  in 
deed,  involved  in  the  exercise  of  that  faculty  by  which 
we  perceive  all  things  as  existing  in  space ;  for  by  this 
faculty  we  place  ourselves  and  other  objects  in  one  com 
mon  space,  and  thus  they  are  exterior  to  us.  It  may  be 
remarked  that  this  apprehension  of  objects  as  external 
to  us,  although  it  assumes  the  idea  of  space,  is  far  from 
being  implied  in  the  idea  of  space.  The  objects  which 
we  contemplate  are  considered  as  existing  in  space,  and 
by  that  means  become  invested  with  certain  mutual  rela 
tions  of  position  ;  but  when  we  consider  them  as  existing 
without  us,  we  make  the  additional  step  of  supposing 
ourselves  and  the  objects  to  exist  in  one  common  space. 
The  question  respecting  the  Ideal  Theory  of  Berkeley  has 
been  mixed  up  with  the  recognition  of  this  condition  of 
the  externality  of  objects.  That  philosopher  maintained, 

OF   THE    IDEA    OF    A    MEDIUM.  281 

as  is  well  known,  that  the  perceptible  qualities  of  bodies 
have  no  existence  except  in  a  perceiving  mind.  This 
system  has  often  been  understood  as  if  he  had  imagined 
the  world  to  be  a  kind  of  optical  illusion,  like  the  images 
which  we  see  when  we  shut  our  eyes,  appearing  to  be 
without  us,  though  they  are  only  in  our  organs ;  and 
thus  this  Ideal  System  has  been  opposed  to  a  belief  in 
an  external  world.  In  truth,  however,  no  such  opposi 
tion  exists.  The  Ideal  System  is  an  attempt  to  explain 
the  mental  process  of  perception,  and  to  get  over  the 
difficulty  of  mind  being  affected  by  matter.  But  the 
author  of  that  system  did  not  deny  that  objects  were 
perceived  under  the  conditions  of  space  and  mechanical 
causation ; — that  they  were  external  and  material  so  far 
as  those  words  describe  perceptible  qualities.  Berkeley's 
system,  however  visionary  or  erroneous,  did  not  prevent 
his  entertaining  views  as  just,  concerning  optics  or  acous 
tics,  as  if  he  had  held  any  other  doctrine  of  the  nature 
of  perception. 

But  when  Berkeley's  theory  was  understood  as  a 
denial  of  the  existence  of  objects  without  us,  how  was  it 
answered  ?  If  we  examine  the  answers  which  are  given 
by  Reid  and  other  philosophers  to  this  hypothesis,  it  will 
be  found  that  they  amount  to  this :  that  objects  are 
without  us,  since  we  perceive  that  they  are  so ;  that  we 
perceive  them  to  be  external,  by  the  same  act  by  which 
we  perceive  them  to  be  objects.  And  thus,  in  this  stage 
of  philosophical  inquiry,  the  externality  of  objects  is  re 
cognized  as  one  of  the  inevitable  conditions  of  our  per 
ception  of  them  ;  and  hence  the  Idea  of  Externality  is 
adopted  as  one  of  the  necessary  foundations  of  all  rea 
soning  concerning  all  objects  whatever. 

3.  Sensation  by  a  Medium. — Objects,  as  we  have  just 
seen,  are  necessarily  apprehended  as  without  us ;  and  in 
general,  as  removed  from  us  by  a  great  or  small  distance. 


Yet  they  affect  our  bodily  senses ;  and  this  leads  us  ir 
resistibly  to  the  conviction  that  they  are  perceived  by 
means  of  something  intermediate.  Vision,  or  hearing, 
or  smell,  or  the  warmth  of  a  fire,  must  be  communicated 
to  us  by  some  medium  of  sensation.  This  unavoidable 
belief  appears  in  all  attempts,  the  earliest  and  the  latest 
alike,  to  speculate  upon  such  subjects.  Thus,  for  in 
stance,  Aristotle  says  *,  "  Seeing  takes  place  in  virtue  of 
some  action  which  the  sentient  organ  suffers :  now  it 
cannot  suffer  action  from  the  colour  of  the  object  di 
rectly  :  the  only  remaining  possible  case  then  is,  that  it 
is  acted  upon  by  an  intervening  Medium ;  there  must 
then  be  an  intervening  Medium."  "  And  the  same  may 
be  said,"  he  adds,  "  concerning  sounding  and  odorous 
bodies ;  for  these  do  not  produce  sensation  by  touching 
the  sentient  organ,  but  the  intervening  Medium  is  acted 
on  by  the  sound  or  the  smell,  and  the  proper  organ,  by 
the  Medium.. ..In  sound  the  Medium  is  air;  in  smell  we 
have  no  name  for  it."  In  the  sense  of  taste,  the  neces 
sity  of  a  Medium  is  not  at  first  so  obviously  seen,  because 
the  object  tasted  is  brought  into  contact  with  the  organ ; 
but  a  little  attention  convinces  us  that  the  taste  of  a 
solid  body  can  only  be  perceived  when  it  is  conveyed 
in  some  liquid  vehicle.  Till  the  fruit  is  crushed,  and 
till  its  juices  are  pressed  out,  we  do  not  distinguish  its 
flavour.  In  the  case  of  heat,  it  is  still  more  clear  that 
we  are  compelled  to  suppose  some  invisible  fluid,  or 
other  means  of  communication,  between  the  distant  body 
which  warms  us  and  ourselves. 

It  may  appear  to  some  persons  that  the  assumption 
of  an  intermedium  between  the  object  perceived  and  the 
sentient  organ  results  from  the  principles  which  form 
the  basis  of  our  mechanical  reasonings, — that  every 
change  must  have  a  cause,  and  that  bodies  can  act  upon 
*  nc/)'(  Vvx^-  IL  7- 

OF    THE    IDEA    OF   A    MEDIUM.  283 

each  other  only  by  contact.  It  cannot  be  denied  that 
this  principle  does  offer  itself  very  naturally  as  the 
ground  of  our  belief  in  media  of  sensation ;  and  it  appears 
to  be  referred  to  for  this  purpose  by  Aristotle  in  the 
passage  quoted  above.  But  yet  we  cannot  but  ask, 
Does  the  principle,  that  matter  produces  its  effect  by 
contact  only,  manifestly  apply  here  ?  When  we  so  apply 
it,  we  include  sensation  among  the  effects  which  material 
contact  produces ; — a  case  so  different  from  any  merely 
mechanical  effect,  that  the  principle,  so  employed,  ap 
pears  to  acquire  a  new  signification.  May  we  not,  then, 
rather  say  that  we  have  here  a  new  axiom, — That  sensa 
tion  implies  a  material  cause  immediately  acting  on  the 
organ, — than  a  new  application  of  our  former  proposi 
tion, — That  all  mechanical  change  implies  contact  ? 

The  solution  of  this  doubt  is  not  of  any  material  con 
sequence  to  our  reasonings ;  for  whatever  be  the  ground 
of  the  assumption,  it  is  certain  that  we  do  assume  the 
existence  of  media  by  which  the  sensations  of  sight, 
hearing,  and  the  like,  are  produced ;  and  it  will  be  seen 
shortly  that  principles  inseparably  connected  with  this 
assumption  are  the  basis  of  the  sciences  now  before  us. 

This  assumption  makes  its  appearance  in  the  physical 
doctrines  of  all  the  schools  of  philosophy.  It  is  ex 
hibited  perhaps  most  prominently  in  the  tenets  of  the 
Epicureans,  who  were  materialists,  and  extended  to  all 
kinds  of  causation  the  axiom  of  the  existence  of  a  cor 
poreal  mechanism  by  which  alone  the  effect  is  produced. 
Thus,  according  to  them,  vision  is  produced  by  certain 
images  or  material  films  which  flow  from  the  object, 
strike  upon  the  eyes,  and  so  become  sensible.  This 
opinion  is  urged  with  great  detail  and  earnestness  by 
Lucretius,  the  poetical  expositor  of  the  Epicurean  creed 
among  the  Romans.  His  fundamental  conviction  of  the 
necessity  of  a  material  medium  is  obviously  the  basis  of 


his  reasoning,  though  he  attempts  to  show  the  existence 
of  such  a  medium  by  facts.  Thus  he  argues'-,  that  by 
shouting  loud  we  make  the  throat  sore  ;  which  shows, 
he  says,  that  the  voice  must  be  material,  so  that  it  can 
hurt  the  passage  in  coming  out. 

Hand  igitur  dubium  est  quin  voces  verbaque  constcnt 
Corporeis  e  principiis  lit  laxlere  possint 

4.  The  Process  of  Perception  of  Secondary  Quali 
ties, — The  likenesses  or  representatives  of  objects  by 
which  they  affect  our  senses  were  called  by  some  writers 
species,  or  sensible  species,  a  term  which  continued  in 
use  till  the  revival  of  science.  It  may  be  observed  that 
the  conception  of  these  species  as  films  cast  off  from  the 
object,  and  retaining  its  shape,  was  different,  as  we  have 
seen,  from  the  view  which  Aristotle  took,  though  it  has 
sometimes  been  called  the  Peripatetic  doctrine  f.  We 
may  add  that  the  expression  was  latterly  applied  to 
express  the  supposition  of  an  emanation  of  any  kind,  and 
implied  little  more  than  that  supposition  of  a  medium 
of  which  we  are  now  speaking.  Thus  Bacon,  after  re 
viewing  the  phenomena  of  sound,  says  \,  "  Videntur 
motus  soni  fieri  per  species  spirituales :  ita  enim  loquen- 
dum  donee  certius  quippiam  inveniatur." 

Though  the  fundamental  principles  of  several  sciences 
depend  upon  the  assumption  of  a  medium  of  perception, 
these  principles  do  not  at  all  depend  upon  any  special 
view  of  the  process  of  our  perceptions.  The  mechanism 
of  that  process  is  a  curious  subject  of  consideration ;  but 
it  belongs  to  physiology,  more  properly  than  either  to 
metaphysics,  or  to  those  branches  of  physics  of  which  we 
are  now  speaking.  The  general  nature  of  the  process  is 
the  same  for  all  the  senses.  The  object  affects  the  ap 
propriate  intermedium  ;  the  medium,  through  the  proper 

*  Lib.  iv.  529  t  Brown,  Vol.  11.  p.  98. 

J  Hist.  Son.  ct  And.,  Vol.  ix.  p.  87. 

OF   THE    IDEA    OF    A    MEDIUM. 

organ,  the  eye,  the  ear,  the  nose,  affects  the  nerves  of 
the  particular  sense ;  and,  by  these,  in  some  way,  the 
sensation  is  conveyed  to  the  mind.  But  to  treat  the 
impression  upon  the  nerves  as  the  act  of  sensation  which 
we  have  to  consider,  would  be  to  mistake  our  object, 
which  is  not  the  constitution  of  the  human  body,  but  of 
the  human  mind.  It  would  be  to  mistake  one  link  for 
the  power  which  holds  the  end  of  the  chain.  No  anato 
mical  analysis  of  the  corporeal  conditions  of  vision,  or 
hearing,  or  feeling  warm,  is  necessary  to  the  sciences  of 
Optics,  or  Acoustics,  or  Thermotics. 

Not  only  is  this  physiological  research  an  extraneous 
part  of  our  subject,  but  a  partial  pursuit  of  such  a 
research  may  mislead  the  inquirer.  We  perceive  objects 
by  means  of  certain  media,  and  by  means  of  certain 
impressions  on  the  nerves :  but  we  cannot  with  pro 
priety  say  that  we  perceive  either  the  media  or  the 
impressions  on  the  nerves.  What  person  in  the  act  of 
seeing  is  conscious  of  the  little  coloured  spaces  on  the 
retina?  or  of  the  motions  of  the  bones  of  the  auditory 
apparatus  whilst  he  is  hearing?  Surely,  no  one.  This 
may  appear  obvious  enough,  and  yet  a  writer  of  no 
common  acuteness,  Dr.  Brown,  has  put  forth  several 
very  strange  opinions,  all  resting  upon  the  doctrine  that 
the  coloured  spaces  on  the  retina  are  the  objects  which 
we  perceive;  and  there  are  some  supposed  difficulties 
and  paradoxes  on  the  same  subject  which  have  become 
quite  celebrated  (as  upright  vision  with  inverted  images), 
arising  from  the  same  confusion  of  thought. 

As  the  consideration  of  the  difficulties  which  have 
arisen  respecting  the  philosophy  of  perception  may  serve 
still  further  to  illustrate  the  principles  on  which  we 
necessarily  reason  respecting  the  secondary  qualities  of 
bodies,  I  shall  here  devote  a  few  pages  to  that  subject. 




1.  WE  cannot  doubt  that  we  perceive  all  secondary 
qualities    by   means    of    immediate    impressions   made, 
through   the    proper   medium    of  sensation,   upon    our 
organs.     Hence  all   the  senses  are  sometimes  vaguely 
spoken  of  as  modifications  of  the  sense  of  feeling.     It 
will,  however,  be  seen,  on  reflection,  that  this  mode  of 
speaking  identifies  in  words  things  which  in  our  concep 
tions  have  nothing  in  common.     No  impression  on  the 
organs  of  touch  can  be  conceived  as  having  any  resem 
blance  to  colour  or  smell.     No  effort,  no  ingenuity,  can 
enable  us  to  describe  the  impressions  of  one  sense  in 
terms  borrowed  from  another. 

The  senses  have,  however,  each  its  peculiar  powers, 
and  these  powers  may  be  in  some  respects  compared,  so 
as  to  show  their  leading  resemblances  and  differences, 
and  the  characteristic  privileges  and  laws  of  each.  This 
is  what  we  shall  do  as  briefly  as  possible. 

SECT.  I. — Prerogatives  of  Sight. 

THE  sight  distinguishes  colours,  as  the  hearing  distin 
guishes  tones ;  the  sight  estimates  degrees  of  brightness, 
the  ear,  degrees  of  loudness ;  but  with  several  resem 
blances,  there  are  most  remarkable  differences  between 
these  two  senses. 

2.  Position. — The  sight  has  this  peculiar  prerogative, 
that  it  apprehends  the  place  of  its  objects  directly  and 
primarily.     We  see  where  an  object  is  at  the  same  in 
stant  that  we  see  what  it  is.     If  we  see  two  objects,  we 
see  their  relative  position.     We  cannot  help  perceiving 


that  one  is  above  or  below,  to  the  right  or  to  the  left  of 
the  other,  if  we  perceive  them  at  all. 

There  is  nothing  corresponding  to  this  in  sound. 
When  we  hear  a  noise,  we  do  not  necessarily  assign  a 
place  to  it.  It  may  easily  happen  that  we  cannot  tell 
from  which  side  a  thunder-clap  comes.  And  though  we 
often  can  judge  in  what  direction  a  voice  is  heard,  this  is 
a  matter  of  secondary  impression,  and  of  inference  from 
concomitant  circumstances,  not  a  primary  fact  of  sensa 
tion.  The  judgments  which  we  form  concerning  the 
position  of  sounding  bodies  are  obtained  by  the  con 
scious  or  unconscious  comparison  of  the  impressions 
made  on  the  two  ears,  and  on  the  bones  of  the  head  in 
general ;  they  are  not  inseparable  conditions  of  hearing. 
We  may  hear  sounds,  and  be  uncertain  whether  they  are 
"  above,  around,  or  underneath !"  but  the  moment  any 
thing  visible  appears,  however  unexpected,  we  can  say, 
"  see  where  it  comes  !" 

Since  we  can  see  the  relative  position  of  things,  we 
can  see  figure,  which  is  but  the  relative  position  of  the 
different  parts  of  the  boundary  of  the  object.  And  thus 
the  whole  visible  world  exhibits  to  us  a  scene  of  various 
shapes,  coloured  and  shaded  according  to  their  form  and 
position,  but  each  having  relations  of  position  to  all  the 
rest ;  and  altogether,  entirely  filling  up  the  whole  range 
which  the  eye  can  command. 

3.  Distance. — The  distance  of  objects  from  us  is  no 
matter  of  immediate  perception,  but  is  a  judgment  and 
inference  formed  from  our  sensations,  in  the  same  way 
as  our  judgment  of  position  by  the  ear.  That  this  is  so, 
was  most  distinctly  shown  by  Berkeley,  in  his  New 
Theory  of  Vision.  The  elements  on  which  we  form  our 
judgment  are,  the  effort  by  which  we  fix  both  eyes  on 
the  same  object,  the  effort  by  which  we  adjust  each  eye 
to  distinct  vision,  and  the  known  forms,  colours,  and 


parts  of  objects,  as  compared  with  their  appearance. 
The  right  interpretation  of  the  information  which  these 
circumstances  give  us  respecting  the  true  distances  and 
forms  of  things,  is  gradually  learnt  by  experience,  the 
lesson  being  begun  in  our  earliest  infancy,  and  incul 
cated  upon  us  every  hour  during  which  we  use  our  eyes. 
The  completeness  with  which  the  lesson  is  learnt  is 
truly  admirable ;  for  we  forget  that  our  conclusion  is 
obtained  indirectly,  and  mistake  a  judgment  on  evidence 
for  an  intuitive  perception.  This,  however,  is  not  more 
surprizing  than  the  rapidity  and  unconsciousness  of  effort 
with  which  we  understand  the  meaning  of  the  speech 
that  we  hear,  or  the  book  that  we  read.  In  both  cases, 
the  habit  of  interpretation  is  become  as  familiar  as  the 
act  of  perception.  And  this  is  the  case  with  regard  to 
vision.  We  see  the  breadth  of  the  street  as  clearly  and 
readily  as  we  see  the  house  on  the  other  side  of  it.  We 
see  the  house  to  be  square,  however  obliquely  it  be  pre 
sented  to  us.  Indeed  the  difficulty  is,  to  recover  the 
consciousness  of  our  real  and  original  sensations ; — to 
discover  what  is  the  apparent  relation  of  the  lines  which 
appear  before  us.  As  we  have  already  said,  in  the  com 
mon  process  of  vision  we  suppose  ourselves  to  see  that 
which  cannot  be  seen ;  and  when  we  would  make  a 
picture  of  an  object,  the  difficulty  is  to  represent  what  is 
visible  and  no  more. 

But  perfect  as  is  our  habit  of  interpreting  what  we 
perceive,  we  could  not  interpret  if  we  did  not  perceive. 
If  the  eye  did  not  apprehend  visible  position,  it  could 
not  infer  actual  position,  which  is  collected  from  visible 
position  as  a  consequence :  if  we  did  not  see  apparent 
figure,  we  could  not  arrive  at  any  opinion  concerning 
real  form.  The  perception  of  place,  which  is  the  prero 
gative  of  the  eye,  is  the  basis  of  all  its  other  superiority. 

The  precision  with  which  the  eye  can  judge  of  appa- 


rent  position  is  remarkable.  If  we  had  before  us  two 
stars  distant  from  each  other  by  one-twentieth  of  the 
moon's  diameter,  we  could  easily  decide  the  apparent 
direction  of  the  one  from  the  other,  as  above  or  below, 
to  the  right  or  left.  Yet  eight  millions  of  stars  might  be 
placed  in  the  visible  hemisphere  of  the  sky  at  such  dis 
tances  from  each  other ;  and  thus  the  eye  would  recog 
nize  the  relative  position  in  a  portion  of  its  range  not 
greater  than  one  eight-millionth  of  the  whole.  Such  is 
the  accuracy  of  the  sense  of  vision  in  this  respect ;  and, 
indeed,  we  might  with  truth  have  stated  it  much  higher. 
Our  judgment  of  the  position  of  distant  objects  in  a 
landscape  depends  upon  features  far  more  minute  than 
the  magnitude  we  have  here  described. 

As  our  object  is  to  point  out  principally  the  differ 
ences  of  the  senses,  we  do  not  dwell  upon  the  delicacy 
with  which  we  distinguish  tints  and  shades,  but  proceed 
to  another  sense. 

SECT.  II. — Prerogatives  of  Hearing. 

THE  sense  of  hearing  has  two  remarkable  prerogatives ; 
it  can  perceive  a  definite  and  peculiar  relation  between 
certain  tones,  and  it  can  clearly  perceive  two  tones  to 
gether;  in  both  these  circumstances  it  is  distinguished 
from  vision,  and  from  the  other  senses. 

4.  Musical  Internals. — We  perceive  that  two  tones 
have,  or  have  not,  certain  definite  relations  to  each 
other,  which  we  call  Concords :  one  sound  is  a  Fifth,  an 
Octave,  &c.,  above  the  other.  And  when  this  is  the  case, 
our  perception  of  the  relation  is  extremely  precise.  It 
is  easy  to  perceive  when  a  fifth  is  out  of  tune  by  one- 
twentieth  of  a  tone  ;  that  is,  by  one-seventieth  of  itself. 
To  this  there  is  nothing  analogous  in  vision.  Colours 
have  certain  vague  relations  to  one  another;  they  look 
well  together,  by  contrast  or  by  resemblance  ;  but 
VOL.  i.  \v.  P.  LT 


is  an  indefinite,  and  in  most  cases  a  casual  and  variable 
feeling.  The  relation  of  complementary  colours  to  one 
another,  as  of  red  to  green,  is  somewhat  more  definite ; 
but  still,  has  nothing  of  the  exactness  and  peculiarity 
which  belongs  to  a  musical  concord.  In  the  case  of  the 
two  sounds,  there  is  an  exact  point  at  which  the  relation 
obtains ;  when  by  altering  one  note  we  pass  this  point, 
the  concord  does  not  gradually  fade  away,  but  instantly 
becomes  a  discord ;  and  if  we  go  further  still,  we  obtain 
another  concord  of  quite  a  different  character. 

We  learn  from  the  theory  of  sound  that  concords 
occur  when  the  times  of  vibration  of  the  notes  have 
exact  simple  ratios;  an  octave  has  these  times  as  1  to  2; 
a  fifth,  as  2  to  3.  According  to  the  undulatory  theory 
of  light,  such  ratios  occur  in  colours,  yet  the  eye  is  not 
affected  by  them  in  any  peculiar  way.  The  times  of  the 
undulations  of  certain  red  and  certain  violet  rays  are 
as  2  to  3,  but  we  do  not  perceive  any  peculiar  harmony 
or  connexion  between  those  colours. 

5.  Chords. — Again,  the  ear  has  this  prerogative,  that 
it  can  apprehend  two  notes  together,  yet  distinct.  If 
two  notes,  distant  by  a  fifth  from  each  other,  are  sounded 
on  two  wind  instruments,  both  they  and  their  musical 
relation  are  clearly  perceived.  There  is  not  a  mixture, 
but  a  concord,  an  interval.  In  colours,  the  case  is  other 
wise.  If  blue  and  yellow  fall  on  the  same  spot,  they 
form  green ;  the  colour  is  simple  to  the  eye ;  it  can  no 
more  be  decomposed  by  the  vision  than  if  it  were  the 
simple  green  of  the  prismatic  spectrum :  it  is  impossible 
for  us,  by  sight,  to  tell  whether  it  is  so  or  not. 

These  are  very  remarkable  differences  of  the  two 
senses :  two  colours  can  be  compounded  into  an  appa 
rently  simple  one  ;  two  sounds  cannot :  colours  pass  into 
each  other  by  gradations  and  intermediate  tints ;  sounds 
pass  from  one  concord  to  another  by  no  gradations  :  the 


most  intolerable  discord  is  that  which  is  near  a  concord. 
We  shall  hereafter  see  how  these  differences  affect  the 
scales  of  sound  and  of  colour. 

6.  Rhythm. — We  might  remark,  that  as  we  see  ob 
jects  in  space,  we  hear  sounds  in  time ;  and  that  we  thus 
introduce  an  arrangement  among  sounds  which  has 
several  analogies  with  the  arrangement  of  objects  in 
space.  But  the  conception  of  time  does  not  seem  to  be 
peculiarly  connected  with  the  sense  of  hearing;  a  faculty 
of  apprehending  tone  and  time,  or  in  musical  phrase 
ology  tune  and  rhythm,  are  certainly  very  distinct.  I 
shall  not,  therefore,  here  dwell  upon  such  analogies. 

The  other  Senses  have  not  any  peculiar  prerogatives, 
at  least  none  which  bear  on  the  formation  of  science.  I 
may,  however,  notice,  in  the  feeling  of  heat,  this  cir 
cumstance  ;  that  it  presents  us  with  two  opposites,  heat 
and  cold,  which  graduate  into  each  other.  This  is  not 
quite  peculiar,  for  vision  also  exhibits  to  us  white  and 
black,  which  are  clearly  opposites,  and  which  pass  into 
each  other  by  the  shades  of  gray. 

SECT.  III. — The  Paradoxes  of  Vision. 

1.  First  Paradox  of  Vision.  Upright  Vision.— 
All  our  senses  appear  to  have  this  in  common ; — That 
they  act  by  means  of  organs,  in  which  a  bundle  of  nerves 
receives  the  impression  of  the  appropriate  medium  of 
the  sense.  In  the  construction  of  these  organs  there  are 
great  differences  and  peculiarities,  corresponding,  in  part 
at  least,  to  the  differences  in  the  information  given. 
Moreover,  in  some  cases,  as  we  have  noted  in  the  case  of 
audible  position  and  visible  distance,  that  which  seems 
to  be  a  perception  is  really  a  judgment  founded  on  per 
ceptions  of  which  we  are  not  directly  aware.  It  will  be 
seen,  therefore,  that  with  respect  to  the  peculiar  powers 
of  each  sense,  it  may  be  asked ; — whether  they  can  be 

u  2 


explained  by  the  construction  of  the  peculiar  organ  ;— 
whether  they  are  acquired  judgments  arid  not  direct 
perceptions  ; — or  whether  they  are  inexplicable  in  either 
of  these  ways,  and  cannot,  at  present  at  least,  be  re 
solved  into  anything  but  conditions  of  the  intellectual 
act  of  perception. 

Two  of  these  questions  with  regard  to  vision,  have 
been  much  discussed  by  psychological  writers:  the  cause 
of  our  seeing  objects  upright  by  inverted  images  on 
the  retina;  and  of  our  seeing  single  with  two  such 

Physiologists  have  very  completely  explained  the 
exquisitely  beautiful  mechanism  of  the  eye,  considered 
as  analogous  to  an  optical  instrument ;  and  it  is  in 
disputable  that  by  means  of  certain  transparent  lenses 
and  humours,  an  inverted  image  of  the  objects  which 
are  looked  at  is  formed  upon  the  retina,  or  fine  net 
work  of  nerve,  with  which  the  back  of  the  eye  is  lined. 
We  cannot  doubt  that  the  impression  thus  produced  on 
these  nerves  is  essential  to  the  act  of  vision ;  and  so  far 
as  we  consider  the  nerves  themselves  to  feel  or  perceive 
by  contact,  we  may  say  that  they  perceive  this  image, 
or  the  affections  of  light  which  it  indicates.  But  we 
cannot  with  any  propriety  say  that  me  perceive,  or  that 
our  mind  perceives,  this  image;  for  we  are  not  conscious 
of  it,  and  none  but  anatomists  are  aware  of  its  existence: 
we  perceive  by  means  of  it. 

A  difficulty  has  been  raised,  and  dwelt  upon  in  a 
most  unaccountable  manner,  arising  from  the  neglect  of 
this  obvious  distinction.  It  has  been  asked,  how  is  it 
that  we  see  an  object,  a  man  for  instance,  upright,  when 
the  immediate  object  of  our  sensation,  the  image  of  the 
man  on  our  retina,  is  inverted  ?  To  this  we  must  answer, 
that  we  see  him  upright  because  the  image  is  inverted ; 
that  the  inverted  image  is  the  necessary  means  of  seeing 


an  upright  object.  This  is  granted,  and  where  then  is 
the  difficulty?  Perhaps  it  may  be  put  thus:  How  is  it 
that  we  do  not  judge  the  man  to  be  inverted,  since  the 
sensible  image  is  so  ?  To  this  we  may  reply,  that  we 
have  no  notion  of  upright  or  inverted,  except  that. which 
is  founded  on  experience,  and  that  all  our  experience, 
without  exception,  must  have  taught  us  that  such  a 
sensible  image  belongs  to  a  man  who  is  in  an  upright 
position.  Indeed,  the  contrary  judgment  is  not  con 
ceivable  ;  a  man  is  upright  whose  head  is  upwards  and 
his  feet  downwards.  But  what  are  the  sensible  images 
of  upwards  asiddonmivardsf'  Whatever  be  our  standard 
of  up  and  down,  the  sensible  representation  of  up  will  be 
an  image  moving  on  the  retina  towards  the  lower  side, 
and  the  sensible  representation  of  down  will  be  a  motion 
towards  the  upper  side.  The  head  of  the  man's  image  is 
towards  the  image  of  the  sky,  its  feet  are  towards  the 
image  of  the  ground ;  how  then  should  it  appear  other 
wise  than  upright  ?  Do  we  expect  that  the  whole  world 
should  appear  inverted  ?  Be  it  so  :  but  if  the  whole  be 
inverted,  how  is  the  relation  of  the  parts  altered  ?  Do 
we  expect  that  we  should  think  our  own  persons  in  par 
ticular  inverted  ?  This  cannot  be,  for  we  look  at  them 
as  we  do  at  other  objects.  Do  we  expect  that  things 
should  appear  to  fall  upwards  ?  Surely  not.  For  what 
do  we  know  of  upwards,  except  that  it  is  the  direction 
in  which  bodies  do  not  fall?  In  short,  the  whole  of 
this  difficulty,  though  it  has  in  no  small  degree  embar 
rassed  metaphysicians,  appears  to  result  from  a  very 
palpable  confusion  of  ideas ;  from  an  attempt  at  com 
parison  of  what  we  see,  with  that  which  the  retina  feels, 
as  if  they  were  separately  presentable.  It  is  a  sufficient 
explanation  to  say,  that  we  do  not  see  the  image  on  the 
retina,  but  see  by  means  of  it.  The  perplexity  does  not 
require  much  more  skill  to  disentangle,  than  it  does 


to  see -that  a  word  written   in  black  ink,  may  signify 


8.  Second  Paradox  of  Vision.  Single  Vision. — 
(1.)  Small  or  Distant  Objects. — The  other  difficulty,  why 
with  two  images  on  the  retina  we  see  only  one  object, 
is  of  a  much  more  real  and  important  kind.  This  effect 
is  manifestly  limited  by  certain  circumstances  of  a  very 
precise  nature ;  for  if  we  direct  our  eyes  at  an  object 
which  is  very  near  the  eye,  we  see  all  other  objects 
double.  The  fact  is  not,  therefore,  that  we  are  incapable 
of  receiving  two  impressions  from  the  two  images,  but 
that,  under  certain  conditions,  the  two  impressions  form 
one.  A  little  attention  shows  us  that  these  conditions 
are,  that  with  both  eyes  we  should  look  at  the  same 
object ;  and  again,  we  find  that  to  look  at  an  object  with 
either  eye,  is  to  direct  the  eye  so  that  the  image  falls 

*  The  explanation  of  our  seeing  objects  erect  when  the  image  is 
inverted  lias  been  put  very  simply,  by  saying,  "  We  call  that  the  lower 
end  of  an  object  which  is  next  the  ground."  The  observer  cannot  look 
into  his  own  eye ;  he  knows  by  experience  what  kind  of  image  cor 
responds  to  a  man  in  an  upright  position.  The  anatomist  tells  him  that 
this  image  is  inverted :  but  this  does  not  disturb  the  process  of  judging 
by  experience.  It  does  not  appear  why  any  one  should  be  perplexed  at 
the  notion  of  seeing  objects  erect  by  means  of  inverted  images,  rather 
than  at  the  notion  of  seeing  objects  large  by  means  of  small  images ;  or 
cubical  and  pyramidal,  by  means  of  images  on  a  spherical  surface ;  or 
green  and  red,  by  means  of  images  on  a  black  surface.  Indeed  some 
persons  have  contrived  to  perplex  themselves  with  these  latter  questions, 
as  well  as  the  first. 

The  above  explanation  is  not  at  all  affected,  as  to  its  substance,  if  we 
adopt  Sir  David  Brewstcr's  expression,  and  say  that  the  line  of  visible 
direction  is  a  line  passing  through  the  center  of  the  spherical  surface  of 
the  retina,  and  therefore  of  course  perpendicular  to  the  surface.  In 
speaking  of  ':  the  inverted  image,"  it  has  always  been  supposed  to  be 
determined  by  such  lines ;  and  though  the  point  where  they  intersect 
may  not  have  been  ascertained  with  exactness  by  previous  physiologists, 
the  philosophical  view  of  the  matter  was  not  in  any  degree  vitiated 
by  this  imperfection. 


on  or  near  a  particular  point  about  the  middle  of  the 
retina.  Thus  these  middle  points  in  the  two  retinas 
correspond,  and  we  see  an  image  single  when  the  two 
images  fall  on  the  corresponding  points. 

Again,  as  each  eye  judges  of  position,  and  as  the  two 
eyes  judge  similarly,  an  object  will  be  seen  in  the  same 
place  by  one  eye  and  by  the  other,  when  the  two  images 
which  it  produces  are  similarly  situated  with  regard  to 
the  corresponding  points  of  the  retina*. 

This  is  the  Law  of  Single  Vision,  at  least  so  far  as 
regards  small  objects;  namely,  objects  so  small  that  in 
contemplating  them  we  consider  their  position  only,  and 
not  their  solid  dimensions.  Single  vision  in  such  cases 
is  a  result  of  the  law  of  vision  simply :  and  it  is  a 
mistake  to  call  in,  as  some  have  done,  the  influence  of 

*  The  explanation  of  single  vision  with  two  eyes  may  be  put  in 
another  form.  Each  eye  judges  immediately  of  the  relative  position  of 
all  objects  within  the  field  of  its  direct  vision.  Therefore  when  we  look 
with  both  eyes  at  a  distant  prospect  (so  distant  that  the  distance 
between  the  eyes  is  small  in  comparison)  the  two  prospects,  being  simi 
lar  collections  of  forms,  will  coincide  altogether,  if  a  corresponding  point 
m  one  and  in  the  other  coincide.  If  this  be  the  case,  the  two  images 
of  every  object  will  fall  upon  corresponding  points  of  the  retina,  and 
will  appear  single. 

If  the  two  prospects  seen  by  the  two  eyes  do  not  exactly  coincide, 
in  consequence  of  nearness  of  the  objects,  or  distortion  of  the  eyes,  but 
if  they  nearly  coincide,  the  stronger  image  of  an  object  absorbs  the 
weaker,  and  the  object  is  seen  single ;  yet  modified  by  the  combination, 
as  will  be  seen  when  we  speak  of  the  siflgle  vision  of  near  objects. 
When  the  two  images  of  an  object  are  considerably  apart,  we  see  it 

This  explanation  is  not  different  in  substance  from  the  one  given  in 
the  text ;  but  perhaps  it  is  better  to  avoid  the  assertion  that  the  law  of 
corresponding  points  is  "  a  distinct  and  original  principle  of  our  consti 
tution,"  as  I  had  stated  in  the  first  edition.  The  simpler  mode  of 
stating  the  law  of  OUT  constitution  appears  to  be  to  say,  that  each  eye 
determines  similarly  the  position  of  objects  ;  and  that  when  the  positions 
of  an  object,  as  seen  by  the  two  eyes,  coincide  (or  nearly  coincide)  the 
object  is  seen  single. 


habit  and  of  acquired  judgments,  in  order  to  determine 
the  result  in  such  cases. 

To  ascribe  the  apparent  singleness  of  objects  to  the 
impressions  of  vision  corrected  by  the  experience  of 
touch'",  would  be  to  assert  that  a  person  who  had  not 
been  in  the  habit  of  handling  what  he  saw,  would  see  all 
objects  double ;  and  also,  to  assert  that  a  person  begin 
ning1  with  the  double  world  which  vision  thus  offers  to 


him,  would,  by  the  continued  habit  of  handling  objects, 
gradually  and  at  last  learn  to  see  them  single.  But 
all  the  facts  of  the  case  show  such  suppositions  to  be 
utterly  fantastical.  No  one  can,  in  this  case,  go  back 
from  the  habitual  judgment  of  the  singleness  of  objects, 
to  the  original  and  direct  perception  of  their  doubleness, 
as  the  draughtsman  goes  back  from  judgments  to  per- 
peption,  in  representing  solid  distances  and  forms  by 
means  of  perspective  pictures.  No  one  can  point  out 
any  case  in  which  the  habit  is  imperfectly  formed ;  even 
children  of  the  most  tender  age  look  at  an  object  with 
both  eyes,  and  see  it  as  one. 

In  cases  when  the  eyes  arc  distorted  (in  squinting), 
one  eye  only  is  used,  or  if  both  are  employed,  there  is 
double  vision ;  and  thus  any  derangement  of  the  corre 
spondence  of  motion  in  the  two  eyes  will  produce  double- 

Brown  is  one  of  those  t  who  assert  that  two  images 
suggest  a  single  object  because  we  have  always  found 
two  images  to  belong  to  a  single  object.  He  urges  as 
an  illustration,  that  the  two  words  "  he  conquered," 
by  custom  excite  exactly  the  same  notion  as  the  one 
Latin  word  "vicit;"  and  thus  that  two  visual  images, 
by  the  effect  of  habit,  produce  the  same  belief  of  a 
single  object  as  one  tactual  impression.  But  in  order 
to  make  this  pretended  illustration  of  any  value,  it  ought 

*  Sec  Brown,  Vol    n.  ]<.  ol  I   Lectures,  Vol.  n.  p.  81. 


to  he  true  that  when  a  person  has  thoroughly  learnt 
the  Latin  language,  he  can  no  longer  distinguish  any 
separate  meaning  in  "  he"  and  in  "  conquered."  We  can 
by  no  effort  perceive  the  double  sensation,  when  we 
look  at  the  object  with  the  two  eyes.  Those  who  squint, 
learn  by  habit  to  see  objects  single:  but  the  habit  which 
they  acquire  is  that  of  attending  to  the  impressions  of 
one  eye  only  at  once,  not  of  combining  the  two  impres 
sions.  It  is  obvious,  that  if  each  eye  spreads  before  us 
the  same  visible  scene,  with  the  same  objects  and  the 
same  relations  of  place,  then,  if  one  object  in  each  scene 
coincide,  the  whole  of  the  two  visible  impressions  will  be 
coincident.  And  here  the  remarkable  circumstance  is, 
that  not  only  each  eye  judges  for  itself  of  the  relations 
of  position  which  come  within  its  field  of  view;  but  that 
there  is  a  superior  and  more  comprehensive  faculty 
which  combines  and  compares  the  two  fields  of  view; 
which  asserts  or  denies  their  coincidence ;  which  con 
templates,  as  in  a  relative  position  to  one  another,  these 
two  visible  worlds,  in  which  all  other  relative  position  is 
given.  This  power  of  confronting  two  sets  of  visible 
images  and  figured  spaces  before  a  purely  intellectual 
tribunal,  is  one  of  the  most  remarkable  circumstances  in 
the  sense  of  vision. 

9.  (2.)  Near  Objects. — We  have  hitherto  spoken 
of  the  singleness  of  objects  whose  images  occupy  corre 
sponding  positions  on  the  retina  of  the  two  eyes.  But 
here  occurs  a  difficulty.  If  an  object  of  moderate  size,  a 
small  thick  book  for  example,  be  held  at  a  little  dis 
tance  from  the  eyes,  it  produces  an  image  on  the  retina 
of  each  eye;  and  these  two  images  are  perspective 
representations  of  the  book  from  different  points  of  view, 
(the  positions  of  the  two  eyes,)  and  are  therefore  of  dif 
ferent  forms.  Hence  the  two  images  cannot  occupy  cor 
responding  points  of  the  retina  throughout  their  whole 


extent.  If  the  central  parts  of  the  two  images  occupy 
corresponding  points,  the  boundaries  of  the  two  will 
not  correspond.  How  is  it  then  consistent  with  the 
law  above  stated,  that  in  this  case  the  object  appears 
single  ? 

It  may  be  observed,  that  the  two  images  in  such  a  case 
will  differ  most  widely  when  the  object  is  not  a  mere  sur 
face,  but  a  solid.  If  a  book,  for  example,  be  held  with 
one  of  its  upright  edges  towards  the  face,  the  right  eye 
will  see  one  side  more  directly  than  the  left  eye,  and 
the  left  eye  will  see  another  side  more  directly,  and  the 
outline  of  the  two  images  upon  the  two  retinas  will  ex 
hibit  this  difference.  And  it  may  be  further  observed, 
that  this  difference  in  the  images  received  by  the  two 
eyes,  is  a  plain  and  demonstrative  evidence  of  the  solidity 
of  the  object  seen ;  since  nothing  but  a  solid  object 
could  (without  some  special  contrivance)  produce  these 
different  forms  of  the  images  in  the  two  eyes. 

Hence  the  absence  of  exact  coincidence  in  the  two 
images  on  the  retina  is  the  necessary  condition  of  the 
solidity  of  the  object  seen,  and  must  be  one  of  the  indi 
cations  by  means  of  which  our  vision  apprehends  an 
object  as  solid.  And  that  this  is  so,  Mr.  Wheatstone 
has  proved  experimentally,  by  means  of  some  most 
ingenious  and  striking  contrivances.  He  has  devised* 
an  instrument  by  which  two  images  (drawn  in  outline) 
differing  exactly  as  much  as  the  two  images  of  a  solid 
body  seen  near  the  face  would  differ,  are  conveyed, 
one  to  one  eye,  and  the  other  to  the  other.  And  it  is 
found  that  when  this  is  effected,  the  object  which  the 
images  represent  is  not  only  seen  single,  but  is  appre 
hended  as  solid  with  a  clearness  and  reality  of  conviction 
<|uitc  distinct  from  any  impression  which  a  mere  per 
spective  representation  can  give. 

;    Phil.  Trans..  1H3H. 


At  the  same  time  it  is  found  that  the  object  is  then 
only  apprehended  as  single  when  the  two  images  are 
such  as  are  capable  of  being  excited  by  one  single  object 
placed  in  solid  space,  and  seen  by  the  two  eyes.  If 
the  images  differ  more  or  otherwise  than  this  condition 
allows,  the  result  is,  that  both  are  seen,  their  lines  cross 
ing  and  interfering  with  one  another. 

It  may  be  observed,  too,  that  if  an  object  be  of  such 
large  size  as  not  to  be  taken  in  by  a  single  glance  of  the 
eyes,  it  is  no  longer  apprehended  as  single  by  a  direct 
act  of  perception ;  but  its  parts  are  looked  at  separately 
and  successively,  and  the  impressions  thus  obtained  are 
put  together  by  a  succeeding  act  of  the  mind.  Hence 
the  objects  which  are  directly  seen  as  solid,  will  be  of 
moderate  size ;  in  which  case  it  is  not  difficult  to  show 
that  the  outlines  of  the  two  images  will  differ  from  each 
other  only  slightly. 

Hence  we  are  led  to  the  following,  as  the  Law  of 
Single  Vision  for  near  objects : — When  the  two  images 
in  the  two  eyes  are  situated  (part  for  part)  nearly,  but 
not  exactly,  upon  corresponding  points,  the  object  is  ap 
prehended  as  single,  if  the  two  images  are  such  as  are 
or  would  be  given  by  a  single  solid  object  seen  by  the 
two  eyes  separately :  and  in  this  case  the  object  is  neces 
sarily  apprehended  as  solid. 

This  law  of  vision  does  not  contradict  that  stated 
above  for  distant  objects  :  for  when  an  object  is  removed 
to  a  considerable  distance,  the  images  in  the  two  eyes 
coincide  exactly,  and  the  object  is  seen  as  single,  though 
without  any  direct  apprehension  of  its  solidity.  The 
first  law  is  a  special  case  of  the  second.  Under  the  con 
dition  of  exactly  corresponding  points,  we  have  the  per 
ception  of  singleness,  but  no  evidence  of  solidity.  Under 
the  condition  of  nearly  corresponding  points,  we  may 
have  the  perception  of  singleness,  and  with  it,  of  solidity. 


We  have  before  noted  it  as  an  important  feature  in 
our  visual  perception,  that  while  we  have  two  distinct 
impressions  upon  the  sense,  which  we  can  contemplate 
separately  and  alternately,  (the  impressions  on  the  two 
eyes,)  we  have  a  higher  perceptive  faculty  which    can 
recognize  these  two  impressions,  exactly  similar  to  each 
other,  as  only  two  images  of  one  and  the  same  assem 
blage  of  objects.     But  we  now  see  that  the  faculty  by 
which   we   perceive  visible  objects  can  do  much  more 
than  this : — it  can  not  only  unite  two  impressions,  and 
recognize  them  as  belonging  to  one  object  in  virtue  of 
their  coincidence,  but  it  can  also  unite  and  identify  them, 
even  when  they  do  not  exactly  coincide.     It  can  correct 
and  adjust  their  small  difference,  so  that  they  are  both 
apprehended  as  representations  of  the  same  figure.     It 
can   infer  from  them  a  real  form,   not  agreeing  with 
either  of  them  ;  and  a  solid  space,  which  they  are  quite 
incapable  of  exemplifying.     The  visual  faculty  decides 
whether  or  not  the  two  ocular  images  can  be  pictures  of 
the  same  solid  object,  and  if  they  can,  it  undoubtingly 
and  necessarily  accepts  them  as  being  so.     This  faculty 
operates  as  if  it  had  the  power  of  calling  before  it  all 
possible  solid  figures,  and  of  ascertaining  by  trial  whether 
any  of  those  will,  at  the  same  time,  fit  both  the  outlines 
which  are  given  by  the  sense.     It  assumes  the  reality 
of  solid  space,  and,  if  it  be  possible,  reconciles  the  appear 
ances  with  that  reality.     And   thus   an  activity  of  the 
mind    of  a  very  remarkable  and  peculiar  kind  is  exer 
cised  in  the  most  common  act  of  seeing. 

10.  It  may  be  said  that  this  doctrine,  of  such  a  visual 
faculty  as  has  been  described,  is  very  vague  and  obscure, 
since  we  arc  not  told  what  are  its  limits.  It  adjusts  and 
corrects  figures  which  nearly  coincide,  so  as  to  identify 
them.  But  lioir  nearly,  it  may  be  asked,  must  the 
figures  approach  oach  other,  in  order  that  this  adjust- 


inent  may  be  possible  *  What  discrepance  renders  im 
possible  the  reconcilement  of  which  we  speak  ?  Is  it 
not  impossible  to  give  a  definite  answer  to  these  ques 
tions,  and  therefore  impossible  to  lay  down  definitely 
such  laws  of  vision  as  we  have  stated  ?  To  this  I  reply, 
that  the  indefiniteness  thus  objected  to  us,  is  no  new 
difficulty,  but  one  with  which  philosophers  are  familiar, 
and  to  which  they  are  already  reconciled.  It  is,  in  fact, 
no  other  than  the  indefiniteness  of  the  limits  of  distinct 
vision.  How  near  to  the  face  must  an  object  be  brought, 
so  that  we  shall  cease  to  see  it  distinctly  ?  The  distance, 
it  will  be  answered,  is  indefinite :  it  is  different  for 
different  persons;  and  for  the  same  person,  it  varies 
with  the  degree  of  effort,  attention,  and  habit.  But  this 
indefiniteness  is  only  the  indefiniteness,  in  another  form, 
of  the  deviation  of  the  two  ocular  images  from  one 
another :  and  in  reply  to  the  question  concerning  them 
we  must  still  say,  as  before,  that  in  doubtful  cases,  the 
power  of  apprehending  an  object  as  single,  when  this 
can  be  done,  will  vary  with  effort,  attention,  and  habit. 
The  assumption  that  the  apparent  object  exists  as  a  real 
figure,  in  real  space,  is  to  be  verified,  if  possible ;  but, 
in  extreme  cases,  from  the  unfitness  of  the  point  of  view, 
or  from  any  other  cause  of  visual  confusion  or  deception, 
the  existence  of  a  real  object  corresponding  to  the  ap 
pearance  may  be  doubtful ;  as  in  any  other  kind  of  per 
ception  it  may  be  doubtful  whether  our  senses,  under 
disadvantageous  circumstances,  give  us  true  information. 
The  vagueness  of  the  limits,  then,  within  which  this 
visual  faculty  can  be  successfully  exercised,  is  no  valid 
argument  against  the  existence  of  the  faculty,  or  the 
truth  of  the  law  which  we  have  stated  concerning  its 


SECT.  IV. — The  Perception  of  Visible  Figure. 

11.  Visible  Figure. — There  is  one  tenet  on  the 
subject  of  vision  which  appears  to  me  so  extravagant 
and  unphilosophical,  that  I  should  not  have  thought  it 
necessary  to  notice  it,  if  it  had  not  been  recently  pro 
mulgated  by  a  writer  of  great  acuteness  in  a  book  which 
has  obtained,  for  a  metaphysical  work,  considerable  cir 
culation.  I  speak  of  Brown's  opinion*  that  we  have  no 
immediate  perception  of  visible  figure.  I  confess  myself 
unable  to  comprehend  fully  the  doctrine  which  he  would 
substitute  in  the  place  of  the  one  commonly  received. 
He  states  it  thus  t:  "When  the  simple  affection  of  sight 
is  blended  with  the  ideas  of  suggestion  [those  arising 
from  touch,  £c.]  in  what  are  termed  the  acquired  per 
ceptions  of  vision,  as,  for  example,  in  the  perception  of 
a  sphere,  it  is  colour  only  which  is  blended  with  the 
large  convexity,  and  not  a  small  coloured  plane."  The 
doctrine  which  Brown  asserts  in  this  and  similar  pas 
sages,  appears  to  be,  that  we  do  not  by  vision  perceive 
both  colour  and  figure > ;  but  that  the  colour  which  we  see 
is  blended  with  the  figure  which  we  learn  the  existence 
of  by  other  means,  as  by  touch.  But  if  this  were  pos 
sible  when  we  can  call  in  other  perceptions,  how  is  it 
possible  when  we  cannot  or  do  not  touch  the  object  ? 
Why  does  the  moon  appear  round,  gibbous,  or  horned  ? 
What  sense  besides  vision  suggests  to  us  the  idea  of  her 
figure  ?  And  even  in  objects  which  we  can  reach,  what 
is  that  circumstance  in  the  sense  of  vision  which  suggests 
to  us  that  the  colour  belongs  to  the  sphere,  except  that 
we  see  the  colour  where  we  see  the  sphere  ?  If  we  do 
not  see  figure,  we  do  not  see  position ;  for  figure  is  the 
relative  position  of  the  parts  of  a  boundary.  If  we  do 
not  see  position,  why  do  we  ascribe  the  yellow  colour  to 

*   Lectures,  Vol.  n.  p.  82.  +   ll>.  Vol.  n.  p.  90. 


the  sphere  on  our  left,  rather  than  to  the  cube  on  our 
right  ?  We  associate  the  colour  with  the  object,  says 
Dr.  Brown ;  but  if  his  opinion  were  true,  we  could  not 
associate  two  colours  with  two  objects,  for  we  could 
not  apprehend  the  colours  as  occupying  two  different 

The  whole  .of  Brown's  reasoning  on  this  subject  is  so 
irreconcileable  with  the  first  facts  of  vision,  that  it  is 
difficult  to  conceive  how  it  could  proceed  from  a  person 
who  has  reasoned  with  great  acuteness  concerning  touch. 
In  order  to  prove  his  assertion,  he  undertakes  to  ex 
amine  the  only  reasons  which,  he  says*,  he  can  imagine 
for  believing  the  immediate  perception  of  visible  figure  : 

(1)  That  it  is  absolutely  impossible,  in  our  present  sen 
sations  of  sight,  to  separate  colour  from  extension  ;  and 

(2)  That  there  are,  in  fact,  figures  on  the  retina  corre 
sponding  to  the  apparent  figures  of  objects. 

On  the  subject  of  the  first  reason,  he  says,  that  the 
figure  which  we  perceive  as  associated  with  colour,  is  the 
real,  and  not  the  apparent  figure.  "  Is  there,"  he  asks, 
"  the  slightest  consciousness  of  a  perception  of  visible 
figure,  corresponding  to  the  affected  portion  of  the 
retina?"  To  which,  though  he  seems  to  think  an  affir 
mative  answer  impossible,  we  cannot  hesitate  to  reply, 
that  there  is  undoubtedly  such  a  consciousness ;  that 
though  obscured  by  being  made  the  ground  of  habitual 
inference  as  to  the  real  figure,  this  consciousness  is  con 
stantly  referred  to  by  the  draughtsman,  and  easily  re 
called  by  any  one.  We  may  separate  colour,  he  says 
again  f,  from  the  figures  on  the  retina,  as  we  may  sepa 
rate  it  from  length,  breadth,  and  thickness,  which  we  do 
not  see.  But  this  is  altogether  false  :  we  cannot  separate 
colour  from  length,  breadth,  and  thickness,  in  any  other 
mty,  than  by  transferring  it  to  the  visible  figure  which 

*  Lectures,  Vol.  n.  p.  $1.  t   //>.  p.  HJ. 


we  do  see.  He  cannot,  ho  allows,  separate  the  colour 
from  the  visible  form  of  the  trunk  of  a  large  oak ;  but 
just  as  little,  he  thinks,  can  he  separate  it  from  the  con 
vex  mass  of  the  trunk,  which  (it  is  allowed  on  all  hands) 
he  does  not  immediately  see.  But  in  this  he  is  mis 
taken  :  for  if  he  were  to  make  a  picture  of  the  oak,  he 
would  separate  the  colour  from  the  convex  shape,  which 
he  does  not  imitate,  but  he  could 'not  separate  it  from 
the  visible  figure,  which  he  does  imitate ;  and  he  would 
then  perceive  that  the  fact  that  he  has  not  an  imme 
diate  perception  of  the  convex  form,  is  necessarily  con 
nected  with  the  fact  that  he  has  an  immediate  percep 
tion  of  the  apparent  figure ;  so  far  is  the  rejection  of 
immediate  perception  in  the  former  case  from  being  a 
reason  for  rejecting  it  in  the  latter. 

Again,  with  regard  to  the  second  argument.  It  does 
not,  he  says,  follow,  that  because  a  certain  figured  por 
tion  of  the  retina  is  affected  by  light,  we  should  see  such 
a  figure  ;  for  if  a  certain  figured  portion  of  the  olfactory 
organ  were  affected  by  odours,  we  should  not  acquire  by 
smell  any  perception  of  such  figure  *.  This  is  merely  to 
say,  that  because  we  do  not  perceive  position  and  figure 
by  one  sense,  we  cannot  do  so  by  another.  But  this 
again  is  altogether  erroneous.  It  is  an  office  of  our 
sight  to  inform  us  of  position,  and  consequently  of 
figure ;  for  this  purpose,  the  organ  is  so  constructed 
that  the  position  of  the  object  determines  the  position 
of  the  point  of  the  retina  affected.  There  is  nothing  of 
this  kind  in  the  organ  of  smell ;  objects  in  different  posi 
tions  and  of  different  forms  do  not  affect  different  parts 
of  the  olfactory  nerve,  or  portions  of  different  shape. 
Different  objects,  remote  from  each  other,  if  perceived 
by  smell,  affect  the  same  part  of  the  olfactory  organs. 
This  is  all  quite  intelligible;  for  it  is  not  the  office  of 

*   Lecluiw,  Vol.  n.  u.  8/. 


smell  to  inform  us  of  position.  Of  what  use  or  meaning 
would  be  the  curious  and  complex  structure  of  the  eye, 
if  it  gave  us  only  such  vague  and  wandering  notions  of 
the  colours  and  forms  of  the  flowers  in  a  garden,  as  we 
receive  from  their  odours  when  we  walk  among  them 
blindfold?  It  is,  as  we  have  said,  the  prerogative  of 
vision  to  apprehend  position :  the  places  of  objects  on 
the  retina  give  this  information.  We  do  not  suppose 
that  the  affection  of  a  certain  shape  of  nervous  expanse 
will  necessarily  and  in  all  cases  give  us  the  impression 
of  figure ;  but  we  know  that  in  vision  it  does ;  and  it  is 
clear  that  if  we  did  not  acquire  our  acquaintance  with 
visible  figure  in  this  way,  we  could  not  acquire  it  in 
any  way*. 

The  whole  of  this  strange  mistake  of  Brown's  appears 
to  arise  from  the  fault  already  noticed ; — that  of  consi 
dering  the  image  on  the  retina  as  the  object  instead  of 
the  means  of  vision.  This  indeed  is  what  he  says :  "  the 
true  object  of  vision  is  not  the  distant  body  itself,  but 
the  light  that  has  reached  the  expansive  termination  of 
the  optic  nerve  f."  Even  if  this  were  so,  we  do  not  see 
why  we  should  not  perceive  the  position  of  the  impres 
sion  on  this  expanded  nerve.  But  as  we  have  already 
said,  the  impression  on  the  nerve  is  the  means  of  vision, 
and  enables  us  to  assign  a  place,  or  at  least  a  direction, 
to  the  object  from  which  the  light  proceeds,  and  thus 
makes  vision  possible.  Brown,  indeed,  pursues  his  own 
peculiar  view  till  he  involves  the  subject  in  utter  contu 
sion.  Thus  he  saysj,  "  According  to  the  common  theory 

*  When  Brown  says  further  (p.  87,)  that  we  can  indeed  show  the 
image  in  the  dissected   eye ;    but  that  "  it  is  not  in  the  dissected  eye 
that  vision  takes  place;"  it  is  difficult  to  see  what  his  drift  is.     Does 
he  doubt  that  there  is  an  imago  formed  in  the  living  as  completely  as 
in  the  dissected  eye  ? 

*  Lectures,  Vol.  11.  p.  ;>7-  +   //>•,  Vol.  n.  p.  89. 
VOL.  I.     W.  P.  X 


[that  figure  can  be  perceived  by  the  eye,]  a  visible 
sphere  is  at  once  to  my  perception  convex  and  plane ; 
and  if  the  sphere  be  a  large  one,  it  is  perceived  at  once 
to  be  a  sphere  of  many  feet  in  diameter,  and  a  plane 
circular  surface  of  the  diameter  of  a  quarter  of  an  inch." 
It  is  easy  to  deduce  these  and  greater  absurdities,  if  we 
proceed  on  his  strange  and  baseless  supposition  that  the 
object  and  the  image  on  the  retina  are  lot//,  perceived. 
But  who  is  conscious  of  the  image  on  the  retina  in  anv 


other  way  than  as  he  sees  the  object  by  means  of  it? 

Brown  seems  to  have  imagined  that  he  was  ana 
lyzing  the  perception  of  figure  in  the  same  manner  in 
which  Berkeley  had  analyzed  the  perception  of  distance. 
lie  ought  to  have  recollected  that  such  an  undertaking, 
to  be  successful,  required  him  to  show  what  elements  he 
analyzed  it  into.  Berkeley  analyzed  the  perception  of 
real  figure  into  the  interpretation  of  visible  figure  accord 
ing  to  certain  rules  which  he  distinctly  stated.  Brown 
analyzes  the  perception  of  visible  figure  into  no  ele 
ments.  Berkeley  says,  that  we  do  not  directly  perceive 
distance,  but  that  we  perceive  something  else,  from 
which  we  infer  distance,  namely,  visible  figure  and  colour, 
and  our  own  efforts  in  seeing;  Brown  says,  that  we  do 
not  see  figure,  but  infer  it ;  what  then  do  we  see,  which 
we  infer  it  from?  To  this  he  offers  no  answer.  He 
asserts  the  seeming  perception  of  visible  figure  to  be  a 
result  of  "  association  ;" — of  "  suggestion."  But  what 
meaning  can  we  attach  to  this?  Suggestion  requires 
something  which  suggests ;  and  not  a  hint  is  given  what 
it  is  which  suggests  position.  Association  implies  two 
things  associated  ;  what  is  the  sensation  which  we  asso 
ciate  with  form  ?  What  is  that  visual  perception  which 
is  not  figure,  and  which  we  mistake  for  figure  ?  What 
perception  is  it  that  suggests  a  square  to  the  eye  ?  What 
impressions  are  those  which  have  been  associated  with 


a  visible  triangle,  so  that  the  revival  of  the  impressions 
revives  the  notion  of  the  triangle  ?  Brown  has  nowhere 
pointed  out  such  perceptions  and  impressions;  nor  indeed 
was  it  possible  for  him  to  do  so ;  for  the  only  visual 
perceptions  which  he  allows  to  remain,  those  of  colour, 
most  assuredly  do  not  suggest  visible  figures  by  their 
differences;  red  is  not  associated  with  square  rather  than 
with  round,  or  with  round  rather  than  square.  On  the 
contrary,  the  eye,  constructed  in  a  very  complex  and 
wonderful  manner  in  order  that  it  may  give  to  us  directly 
the  perception  of  position  as  well  as  of  colour,  has  it  for 
one  of  its  prerogatives  to  give  us  this  information ;  and 
the  perception  of  the  relative  position  of  each  part  of 
the  visible  boundary  of  an  object  constitutes  the  percep 
tion  of  its  apparent  figure ;  which  faculty  we  cannot 
deny  to  the  eye  without  rejecting  the  plain  and  constant 
evidence  of  our  senses,  making  the  mechanism  of  the 
eye  unmeaning,  confounding  the  object  with  the  means 
of  vision,  and  rendering  the  mental  process  of  vision 
utterly  unintelligible. 

Having  sufficiently  discussed  the  processes  of  per 
ception,  I  now  return  to  the  consideration  of  the  Ideas 
which  these  processes  assume. 





1.  IN  what  precedes,  we  have  shown  by  various  con 
siderations  that  we  necessarily  and  universally  assume 
the  perception  of  secondary  qualities  to  take  place  by 
means  of  a  medium  interjacent  between  the  object  and 
the  person  perceiving.  Perception  is  affected  by  various 



peculiarities,  according  to  the  nature  of  the  quality  per 
ceived  :  but  in  all  cases  a  medium  is  equally  essential  to 
the  process. 

This  principle,  which,,  as  we  have  seen,  is  accepted  as 
evident  by  the  common  understanding  of  mankind,  is 
confirmed  by  all  additional  reflection  and  discipline  of 
the  mind,  and  is  the  foundation  of  all  the  theories  which 
have  been  proposed  concerning  the  processes  by  which 
the  perception  takes  place,  and  concerning  the  modifi 
cations  of  the  qualities  thus  perceived.  The  medium,  and 
the  mode  in  which  the  impression  is  conveyed  through 
the  medium,  seem  to  be  different  for  different  qualities ; 
but  the  existence  of  the  medium  leads  to  certain  neces 
sary  conditions  or  alternatives,  which  have  successively 
made  their  appearance  in  science,  in  the  course  of  the 
attempts  of  men  to  theorize  concerning  the  principal 
secondary  qualities,  sound,  light,  and  heat.  We  must 
now  point  out  some  of  the  ways,  at  first  imperfect  and 
erroneous,  in  which  the  consequences  of  the  fundamental 
assumption  were  traced. 

2.  Sound. — In  all  cases  the  medium  of  sensation, 
whatever  it  is,  is  supposed  to  produce  the  effect  of  con 
veying  secondary  qualities  to  our  perception  by  means 
of  its  primary  qualities.  It  was  conceived  to  operate  by 
the  size,  form,  and  motion  of  its  parts.  This  is  a  funda 
mental  principle  of  the  class  of  sciences  of  which  we 
have  at  present  to  speak. 

It  was  assumed  from  the  first,  as  we  have  seen  in  the 
passage  lately  quoted  from  Aristotle*,  that  in  the  con 
veyance  of  sound,  the  medium  of  communication  was 
the  air.  But  although  the  first  theorists  were  right 
so  far,  that  circumstance  did  not  prevent  their  going 
entirely  wrong  when  they  had  further  to  determine  the 
nature  of  the  process.  It  was  conceived  by  Aristotle 

*  Supr.,  p.  282. 


that  the  air  acted  after  the  manner  of  a  rigid  body  ;— 
like  a  staff,  which,  receiving  an  impulse  at  one  end,  trans 
mits  it  to  the  other.  Now  this  is  altogether  an  erro 
neous  view  of  the  manner  in  which  the  air  conveys  the 
impulse  by  which  sound  is  perceived.  An  approach  was 
made  to  the  true  view  of  this  process,  by  assimilating  it 
to  the  diffusion  of  the  little  circular  waves  which  are 
produced  on  the  surface  of  still  water  when  a  stone  is 
dropt  into  it.  These  little  waves  begin  from  the  point 
thus  disturbed,  and  run  outwards,  expanding  on  every 
side,  in  concentric  circles,  till  they  are  lost.  The  propa 
gation  of  sound  through  the  air  from  the  point  where  it 
is  produced,  was  compared  by  Vitruvius  to  this  diffu 
sion  of  circular  waves  in  water;  and  thus  the  notion  of 
a  propagation  of  impulse  by  the  waves  of  a  fluid  was 
introduced,  in  the  place  of  the  former  notion  of  the 
impulse  of  an  unyielding  body. 

But  though,  taking  an  enlarged  view  of  the  nature 
of  the  progress  of  a  wave,  this  is  a  just  representation 
of  the  motion  of  air  in  conveying  sound,  we  cannot  sup 
pose  that  the  process  was,  at  the  period  of  which  we 
speak,  rightly  understood.  For  the  waves  of  water  were 
contemplated  only  as  affecting  the  surface  of  the  water ; 
and  as  the  air  has  no  surface,  the  communication  must 
take  place  by  means  of  an  internal  motion,  which  can 
bear  only  a  remote  and  obscure  resemblance  to  the  waves 
which  we  see.  And  even  with  regard  to  the  waves  of 
water,  the  mechanism  by  which  they  are  produced  and 
transferred  was  not  at  all  understood ;  so  that  the  com 
parison  employed  by  Vitruvius  must  be  considered  rather 
as  a  loose  analogy  than  as  an  exact  scientific  explanation. 

No  correct  account  of  such  motions  was  given,  till 
the  formation  of  the  science  of  Mechanics  in  modern 
times  had  enabled  philosophers  to  understand  more  dis 
tinctly  the  mode  in  which  motion  is  propagated  through 


a  fluid,  and  to  discern  the  forces  which  the  process  calls 
into  play,  so  as  to  continue  the  motion  once  begun. 
Newton  introduced  into  this  subject  the  exact  and  rigor 
ous  conception  of  an  undulation,  which  is  the  true  key  to 
the  explanation  of  impulses  conveyed  through  a  fluid. 

Even  at  the  present  day.  the  right  apprehension  of 
the  nature  of  an  undulation  transmitted  through  a  fluid 
is  found  to  be  very  difficult  for  all  persons  except  those 
whose  minds  have  been  duly  disciplined  by  mathematical 
studies.  When  we  see  a  wave  run  along  the  surface  of 
water,  we  are  apt  to  imagine  at  first  that  a  portion  of 
the  fluid  is  transferred  bodily  from  one  place  to  another. 
But  with  a  little  consideration  we  may  easily  satisfy 
ourselves  that  this  is  not  so :  for  if  we  look  at  a  field  of 
standing  corn,  when  a  breeze  blows  over  it,  we  see  waves 
like  those  of  water  run  along  its  surface.  Yet  it  is  clear 
that  in  this  case  the  separate  stalks  of  corn  only  bend 
backwards  and  forwards,  and  no  portion  of  the  grain  is 
really  conveyed  from  one  part  of  the  field  to  the  other. 
This  is  obvious  even  to  popular  apprehension.  The  poet 
speaks  of 

The  rye, 

That  stoops  its  head  when  whirlwinds  rave 
And  springs  again  in  eddying  wave 
As  each  wild  gust  sweeps  by. 

Each  particle  of  the  mass  in  succession  has  a  small 
motion  backwards  and  forwards ;  and  by  this  means  a 
large  ridge  made  by  many  such  particles  runs  along  the 
mass  to  any  distance.  This  is  the  true  conception  of 
an  undulation  in  general. 

Thus,  when  an  undulation  is  propagated  in  a  fluid, 
it  is  not  matter,  but  form,  which  is  transmitted  from  one 
place  to  another.  The  particles  along  the  line  of  each 
wave  assume  a  certain  arrangement,  and  this  arrange 
ment  passes  from  one  part  to  another,  the  particles 


changing  their  places  only  within  narrow  limits,  so  as  to 
lend  themselves  successively  to  the  arrangements  by 
which  the  successive  waves,  and  the  intervals  between 
the  waves,  are  formed. 

When  such  an  undulation  is  propagated  through 
air,  the  wave  is  composed,  not,  as  in  water,  of  particles 
which  are  higher  than  the  rest,  but  of  particles  which 
are  closer  to  each  other  than  the  rest.  The  wave  is  not 
a  ridge  of  elevation,  but  a  line  of  condensation ;  and  as 
in  water  we  have  alternately  elevated  and  depressed 
lines,  we  have  in  air  lines  alternately  condensed  and 
rarefied.  And  the  motion  of  the  particles  is  not,  as  in 
water,  up  and  down,  in  a  direction  transverse  to  that  of 
the  wave  which  runs  forwards ;  in  the  motion  of  an 
undulation  •through  air  the  motion  of  each  particle  is 
alternately  forwards  and  backwards,  while  the  motion 
of  the  undulation  is  constantly  forwards. 

This  precise  and  detailed  account  of  the  undulatory 
motion  of  air  by  which  sound  is  transmitted  was  first 
given  by  Newton.  He  further  attempted  to  determine 
the  motions  of  the  separate  particles,  and  to  point  out 
the  force  by  which  each  particle  affects  the  next,  so  as 
to  continue  the  progress  of  the  undulation  once  begun. 
The  motions  of  each  particle  must  be  oscillatory;  he 
assumed  the  oscillations  to  be  governed  by  the  simplest 
law  of  oscillation  which  had  come  under  the  notice  of 
mathematicians,  (that  of  small  vibrations  of  a  pendulum;) 
and  he  proved  that  in  this  manner  the  forces  which  are 
called  into  play  by  the  contraction  and  expansion  of  the 
parts  of  the  elastic  fluid  are  such  as  the  continuance  of 
the  motion  requires. 

Newton's  proof  of  the  exact  law  of  oscillatory  motion 
of  the  aerial  particles  was  not  considered  satisfactory  by 
succeeding  mathematicians ;  for  it  was  found  that  the 
same  result,  the  development  of  forces  adequate  to  con- 


tinue  the  motion,  would  follow  if  any  other  law  of  the 
motion  were  assumed.  Cramer  proved  this  by  a  sort  of 
parody  of  Newton's  proof,  in  which,  by  the  alteration  of 
a  few  phrases  in  this  formula  of  demonstration,  it  was 
made  to  establish  an  entirely  different  conclusion. 

But  the  general  conception  of  an  undulation  as  pre 
sented  by  Newton  was,  as  from  its  manifest  mechanical 
truth  it  could  not  fail  to  be,  accepted  by  all  mathemati 
cians  ;  and  in  proportion  as  the  methods  of  calculating 
the  motions  of  fluids  were  further  improved,  the  neces 
sary  consequences  of  this  conception,  in  the  communica 
tion  of  sound  through  air,  were  traced  by  unexceptionable 
reasoning.  This  was  especially  done  by  Euler  and 
Lagrange,  whose  memoirs  on  such  motions  of  fluids  are 
some  of  the  most  admirable  examples  which  exist,  of 
refined  mathematical  methods  applied  to  the  solution  of 
difficult  mechanical  problems. 

But  the  great  step  in  the  formation  of  the  theory  of 
sound  was  undoubtedly  that  which  we  have  noticed,  the 
introduction  of  the  Conception  of  an  Undulation  such  as 
we  have  attempted  to  describe  it: — a  state,  condition,  or 
arrangement  of  the  particles  of  a  fluid,  which  is  trans 
ferred  from  one  part  of  space  to  another  by  means  of 
small  motions  of  the  particles,  altogether  distinct  from 
the  movement  of  the  undulation  itself.  This  is  a  con 
ception  which  is  not  obvious  to  common  apprehension. 
It  appears  paradoxical  at  first  sight  to  speak  of  a  large 
wave  (as  the  tide-wave)  running  up  a  river  at  the  rate  of 
twenty  miles  an  hour,  while  the  stream  of  the  river  is 
all  the  while  flowing  downwards.  Yet  this  is  a  very 
common  fact.  And  the  conception  of  such  a  motion 
must  be  fully  mastered  by  all  who  would  reason  rightly 

*  O  «/ 

concerning  the   transmission  of  impressions  through  a 

We  have  described  the  motion  of  sound  as  produced 


by  small  motions  of  the  particle  forwards  and  backwards, 
while  the  waves,  or  condensed  and  rarefied  lines,  move 
constantly  forwards.  It  may  be  asked  what  right  we 
have  to  suppose  the  motion  to  be  of  this  kind,  since 
when  sound  is  heard,  no  such  motions  of  the  particles  of 
air  can  be  observed,  even  by  refined  methods  of  observa 
tion.  Thus  Bacon  declares  himself  against  the  hypothesis 
of  such  a  vibration,  since,  as  he  remarks,  it  cannot  be 
perceived  in  any  visible  impression  upon  the  flame  of  a 
candle.  And  to  this  we  reply,  that  the  supposition  of 
this  vibration  is  made  in  virtue  of  a  principle  which 
is  involved  in  the  original  assumption  of  a  medium ; 
namely,  That  a  medium,  in  conveying  secondary  quali 
ties,  operates  by  means  of  its  primary  qualities,  the 
bulk,  figure,  motion,  and  other  mechanical  properties  of 
its  parts.  This  is  an  Axiom  belonging  to  the  Idea  of  a 
Medium.  In  virtue  of  this  axiom  it  is  demonstrable  that 
the  motion  of  the  air,  when  any  how  disturbed,  must  be 
such  as  is  supposed  in  our  acoustical  reasonings.  For 
the  elasticity  of  the  parts  of  the  air,  called  into  play  by 
its  expansion  and  contraction,  lead,  by  a  mechanical 
necessity,  to  such  a  motion  as  we  have  described.  We 
may  add  that,  by  proper  contrivances,  this  motion  may 
be  made  perceptible  in  its  visible  effects.  Thus  the 
theory  of  sound,  as  an  impression  conveyed  through  air, 
is  established  upon  evident  general  principles,  although 
the  mathematical  calculations  which  arc  requisite  to 
investigate  its  consequences  are,  some  of  them,  of  a  very 
recondite  kind. 

'>.  Liylit. — The  early  attempts  to  explain  vision 
represented  it  as  performed  by  means  of  material  rays 
proceeding /Wftt  the  eye,  by  the  help  of  which  the  eye 
felt  out  the  form  and  other  visible  qualities  of  an  object, 
as  a  blind  man  might  do  with  his  staff.  lUit  this  opi 
nion  could  not  keep  its  ground  long:  for  it  did  not  even 


explain  the  fact  that  light  is  necessary  to  vision.  Light  as 
a  peculiar  medium  was  next  assumed  as  the  machinery 
of  vision ;  but  the  mode  in  which  the  impression  was 
conveyed  through  the  medium  was  left  undetermined, 
and  no  advance  was  made  towards  sound  theory,  on  that 
subject,  by  the  ancients. 

In  modern  times,  when  the  prevalent  philosophy 
began  to  assume  a  mechanical  turn  (as  in  the  theories 
of  Descartes),  light  was  conceived  to  be  a  material  sub 
stance  which  is  emitted  from  luminous  bodies,  and  which 
is  also  conveyed  from  all  bodies  to  the  eye,  so  as  to 
render  them  visible.  The  various  changes  of  direction 
by  which  the  rays  of  light  are  affected,  (reflexion,  refrac 
tion,  &c.,)  Descartes  explained,  by  considering  the  par 
ticles  of  light  as  small  globules,  which  change  their 
direction  when  they  impinge  upon  other  bodies,  accord 
ing  to  the  laws  of  mechanics.  Newton,  with  a  much 
more  profound  knowledge  of  mechanics  than  Descartes 
possessed,  adopted,  in  the  most  mature  of  his  specula 
tions,  nearly  the  same  view  of  the  nature  of  light ;  and 
endeavoured  to  show  that  reflexion,  refraction,  and  other 
properties  of  light,  might  be  explained  as  the  effects 
which  certain  forces,  emanating  from  the  particles  of 
bodies,  produce  upon  the  luminiferous  globules. 

But  though  some  of  the  properties  of  light  could  thus 
be  accounted  for  by  the  assumption  of  particles  emitted 
from  luminous  bodies,  and  reflected  or  refracted  by  forces, 
other  properties  came  into  view  which  would  not  admit 
of  the  same  explanation.  The  phenomena  of  diffraction 
(the  fringes  which  accompany  shadows)  could  never  be 
truly  represented  by  such  an  hypothesis,  in  spite  of  many 
attempts  which  were  made.  And  the  colours  of  thin 
1>l«t(>s,  which  show  the  rays  of  light  to  be  affected  by  an 
alternation  of  two  different  conditions  at  small  intervals 
along  their  length,  led  Newton  himself  to  incline,  often 


and  strongly,  to  some  hypothesis  of  undulation.  The 
double  refraction  of  Iceland  spar,  a  phenomenon  in  itself 
very  complex,  could,  it  was  found  by  Huyghcns,  be 
expressed  with  great  simplicity  by  a  certain  hypothesis 
of  undulations. 

Two  hypotheses  of  the  nature  of  the  luminiferous 
medium  were  thus  brought  under  consideration  ;  the  one 
representing  Light  as  Matter  emitted  from  the  luminous 
object,  the  other,  as  Undulations  propagated  through  a 
fluid.  These  two  hypotheses  remained  in  presence  of 
each  other  during  the  whole  of  the  last  century,  neither 
of  them  gaining  any  material  advantage  over  the  other, 
though  the  greater  part  of  mathematicians,  following 
Newton,  embraced  the  emission  theory.  But  at  the 
beginning  of  the  present  century,  an  additional  class  of 
phenomena,  those  of  the  interference  of  two  rays  of 
light,  were  brought  under  consideration  by  Dr.  Young; 
and  these  phenomena  were  strongly  in  favour  of  the 
undulatory  theory,  while  they  were  irreconcilable  with 
the  hypothesis  of  emission.  If  it  had  not  been  for  the 
original  bias  of  Newton  and  his  school  to  the  other  side, 
there  can  be  little  doubt  that  from  this  period  light  as 
well  as  sound  would  have  been  supposed  to  be  pro 
pagated  by  undulations;  although  in  this  case  it  was 
necessary  to  assume  as  the  vehicle  of  such  undulations 
a  special  medium  or  ether.  Several  points  of  the  phe 
nomena  of  vision  no  doubt  remained  unexplained  by  the 
undulatory  theory,  as  absorption,  and  the  natural  colours 
of  bodies ;  but  such  facts,  though  they  did  not  confirm, 
did  not  evidently  contradict  the  theory  of  a  luminiferous 
other ;  and  the  facts  which  such  a  theory  did  explain,  it 
explained  with  singular  happiness  and  accuracy. 

But  before  this  undulatory  theory  could  begenerallv 
accepted,  it  was  presented  in  an  entirely  new  point  of 
view  by  being  combined  with  the  facts  of  polarization. 


The  general  idea  of  polarization  must  be  illustrated  here 
after  ;  but  we  may  here  remark  that  Young  and  Fresnel, 
who  had  adopted  the  undulatory  theory,  after  being 
embarrassed  for  some  time  by  the  new  facts  which  were 
thus  presented  to  their  notice,  at  last  saw  that  these 
facts  might  be  explained  by  conceiving  the  vibrations  to 
be  transverse  to  the  ray,  the  motions  of  the  particles 
being  not  backwards  and  forwards  in  the  line  in  which 
the  impulse  travels,  but  to  the  right  and  left  of  that 
line.  This  conception  of  transverse  vibrations,  though 
quite  unforeseen,  had  nothing  in  it  which  was  at  all  diffi 
cult  to  reconcile  with  the  general  notion  of  an  undula 
tion.  We  have  described  an  undulation,  or  wave,  as  a 
certain  condition  or  arrangement  of  the  particles  of  the 
fluid  successively  transferred  from  one  part  of  space  to 
another :  and  it  is  easily  conceivable  that  this  arrange 
ment  or  wave  may  be  -produced  by  a  lateral  transfer  of 
the  particles  from  their  quiescent  positions.  This  con 
ception  of  transverse  vibrations  being  accepted,  it  was 
found  that  the  explanation  of  the  phenomena  of  polari 
zation  and  of  those  of  interference  led  to  the  same 
theory  with  a  correspondence  truly  wonderful ;  and  this 
coincidence  in  the  views,  collected  from  two  quite  dis 
tinct  classes  of  phenomena,  was  justly  considered  as  an 
almost  demonstrative  evidence  of  the  truth  of  this  undu- 
latory  theory. 

It  remained  to  be  considered  whether  the  doctrine 
of  transverse  vibrations  in  a  fluid  could  be  reconciled 
with  the  principles  of  mechanics.  And  it  was  found 
that  by  making  certain  suppositions,  in  which  no  in 
herent  improbability  existed,  the  hypothesis  of  trans 
verse  vibrations  would  explain  the  laws,  both  of  inter 
ference  and  of  polarization  of  light,  in  air  and  in  crystals 
of  all  kinds,  with  a  surprizing  fertility  and  fidelity. 

Thus  the  undulatory  theory  of  light,  like  the  undu- 


latory  theory  of  sound,  is  recommended  by  its  conformity 
to  the  fundamental  principle  of  the  Secondary  Mecha 
nical  Sciences,  that  the  medium  must  be  supposed  to 
transmit  its  peculiar  impulses  according  to  the  laws  of 
mechanics.  Although  no  one  had  previously  dreamt  of 
qualities  being  conveyed  through  a  medium  by  such  a 
process,  yet  when  it  is  once  suggested  as  the  only  mode 
of  explaining  some  of  the  phenomena,  there  is  nothing 
to  prevent  our  accepting  it  entirely,  as  a  satisfactory 
theory  for  all  the  known  laws  of  light. 

4.  Heat. — With  regard  to  heat  as  with  regard  to 
light,  a  fluid  medium  was  necessarily  assumed  as  the 
vehicle  of  the  property.  During  the  last  century,  this 
medium  was  supposed  to  be  an  emitted  fluid.  And 
many  of  the  ascertained  Laws  of  Heat,  those  which 
prevail  with  regard  to  its  radiation  more  especially,  were 
well  explained  by  this  hypothesis*.  Other  effects  of 
heat,  however,  as  for  instance  latent  heat^,  and  the 
change  of  consistence  of  bodies^,  were  not  satisfactorily 
brought  into  connexion  with  the  hypothesis1;  while  con 
duction  §,  which  at  first  did  not  appear  to  result  from 
the  fundamental  assumption,  was  to  a  certain  extent 
explained  as  internal  radiation. 

But  it  was  by  no  means  clear  that  an  undulatory 
theory  of  heat  might  not  be  made  to  explain  these 
phenomena  equally  well.  Several  philosophers  inclined 
to  such  a  theory ;  and  finally,  Ampere  showed  that  the 
doctrine  that  the  heat  of  a  body  consists  in  the  undula 
tions  of  its  particles  propagated  by  means  of  the  undula 
tions  of  a  medium,  might  be  so  adjusted  as  to  explain  all 
which  the  theory  of  emission  could  explain,  and  more 
over  to  account  for  facts  and  laws  which  were  out  of 

*  See  the  Account  of  the  Theory  of  Exchanges,  Hi.ff.  Ind.  Sci., 
B.  x.  c.  i.  sect.  2.  t  76.,  c.  ii.  sect.  3. 

£   76.,  c.  ii.  sect.  (2.  §  7/>.,  c.  i.  sect.  7- 


the  reach  of  that  theory.  About  the  same  time  it  was 
discovered  by  Prof.  Forbes  and  M.  Nobili  that  radiant 
heat  is,  under  certain  circumstances,  polarized.  Now 
polarization  had  been  most  satisfactorily  explained  by 
means  of  transverse  undulations  in  the  case  of  light ; 
while  all  attempts  to  modify  the  emission  theory  so  as  to 
include  polarization  in  it,  had  been  found  ineffectual. 
Hence  this  discovery  was  justly  considered  as  lending 
great  countenance  to  the  opinion  that  heat  consists  in 
the  vibrations  of  its  proper  medium. 

But  what  is  this  medium  ?  Is  it  the  same  by  which 
the  impressions  of  light  are  conveyed  ?  This  is  a  difficult 
question ;  or  rather  it  is  one  which  we  cannot  at  present 
hope  to  answer  with  certainty.  No  doubt  the  con 
nexion  between  light  and  heat  is  so  intimate  and  con 
stant,  that  we  can  hardly  refrain  from  considering  them 
as  affections  of  the  same  medium.  But  instead  of 
attempting  to  erect  our  systems  on  such  loose  and 
general  views  of  connexion,  it  is  rather  the  business  of 
the  philosophers  of  the  present  day  to  determine  the 
laws  of  the  operation  of  heat,  and  its  real  relation  to 
light,  in  order  that  we  may  afterwards  be  able  to  con 
nect  the  theories  of  the  two  qualities.  Perhaps  in  a 
more  advanced  state  of  our  knowledge  we  may  be  able 
to  state  it  as  an  axiom,  that  two  secondary  qualities, 
which  are  intimately  connected  in  their  causes  and 
effects,  must  be  affections  of  the  same  medium.  But  at 
present  it  does  not  appear  safe  to  proceed  upon  such  a 
principle,  although  many  writers,  in  their  speculations 
both  concerning  light  and  heat,  and  concerning  other 
properties,  have  not  hesitated  to  do  so. 

Some  other  consequences  follow  from  the  Idea  of  a 
Medium  which  must  be  the  subject  of  another  chapter. 


SECT.  I. — Scales  of  Qualities  in  general. 

THE  ultimate  object  of  our  investigation  in  each  of  the 
Secondary  Mechanical  Sciences,  is  the  nature  of  the  pro 
cesses  by  which  the  special  impressions  of  sound,  light, 
and  heat,  are  conveyed,  and  the  modifications  of  which 
these  processes  are  susceptible.  And  of  this  investiga 
tion,  as  we  have  seen,  the  necessary  basis  is  the  principle, 
that  these  impressions  are  transmitted  by  means  of  a 
medium.  But  before  we  arrive  at  this  ultimate  object, 
we  may  find  it  necessary  to  occupy  ourselves  with  seve 
ral  intermediate  objects :  before  we  discover  the  cause, 
it  may  be  necessary  to  determine  the  lares  of  the  phe 
nomena.  Even  if  we  cannot  immediately  ascertain  the 
mechanism  of  light  or  heat,  it  may  still  be  interesting 
and  important  to  arrange  and  measure  the  effects  which 
we  observe. 

The  idea  of  a  medium  affects  our  proceeding  in  this 
research  also.  We  cannot  measure  secondary  qualities 
in  the  same  manner  in  which  we  measure  primary  quali 
ties,  by  a  mere  addition  of  parts.  There  is  this  leading 
and  remarkable  difference,  that  while  both  classes  of 
qualities  are  susceptible  of  changes  of  magnitude,  primary 
qualities  increase  by  addition  of  extension,  secondary,  by 
augmentation  of  intensity.  A  space  is  doubled  when 
another  equal  space  is  placed  by  its  side;  one  weight 
joined  to  another  makes  up  the  sum  of  the  two.  But 
when  one  degree  of  warmth  is  combined  with  another, 
or  one  shade  of  red  colour  with  another,  we  cannot  in 
like  manner  talk  of  the  sum.  The  component  parts  do 
not  evidently  retain  their  separate  existence ;  we  cannot 


separate  a  strong  green  colour  into  two  weaker  ones,  as 
we  can  separate  a  large  force  into  two  smaller.  The 
increase  is  absorbed  into  the  previous  amount,  and  is  no 
longer  in  evidence  as  a  part  of  the  whole.  And  this  is 
the  difference  which  has  given  birth  to  the  two  words 
extended,  and  intense.  That  is  extended  which  has 
"partes  extra  paries,"  parts  outside  of  parts:  that  is 
intense  which  becomes  stronger  by  some  indirect  and 
unapparent  increase  of  agency,  like  the  stretching  of  the 
internal  springs  of  a  machine,  as  the  term  intense  im 
plies.  Extended  magnitudes  can  at  will  be  resolved 
into  the  parts  of  which  they  were  originally  composed, 
or  any  other  which  the  nature  of  their  extension  admits ; 
their  proportion  is  apparent ;  they  are  directly  and  at 
once  subject  to  the  relations  of  number.  Intensive 
magnitudes  cannot  be  resolved  into  smaller  magnitudes ; 
we  can  see  that  they  differ,  but  we  cannot  tell  in  what 
proportion ;  we  have  no  direct  measure  of  their  quan 
tity.  How  many  times  hotter  than  blood  is  boiling- 
water?  The  answer  cannot  be  given  by  the  aid  of  our 
feelings  of  heat  alone. 

The  difference,  as  we  have  said,  is  connected  with 
the  fundamental  principle  that  we  do  not  perceive 
secondary  qualities  directly,  but  through  a  medium.  We 
have  no  natural  apprehension  of  light,  or  sound,  or  heat, 
as  they  exist  in  the  bodies  from  which  they  proceed,  but 
only  as  they  affect  our  organs.  We  can  only  measure 
them,  therefore,  by  some  Scale  supplied  by  their  effects. 
And  thus  while  extended  magnitudes,  as  space,  time,  are 
measurable  directly  and  of  themselves;  intensive  mag 
nitudes,  as  brightness,  loudness,  heat,  are  measurable 
only  by  artificial  means  and  conventional  scales.  Space, 
time,  measure  themselves :  the  repetition  of  a  smaller 
space,  or  time,  while  it  composes  a  larger  one,  measures 
it.  But  for  light  and  heat  we  must  have  Photometers 


and  Thermometers,  which  measure  something  which  is 
assumed  to  be  an  indication  of  the  quality  in  question. 
In  one  case,  the  mode  of  applying  the  measure,  and 
the  meaning  of  the  number  resulting,  are  seen  by  intui 
tion  ;  in  the  other,  they  are  consequences  of  assumption 
and  reasoning.  In  the  one  case,  they  are  Units,  of 
which  the  extension  is  made  up ;  in  the  other,  they  are 
Degrees  by  which  the  intensity  ascends. 

2.  When  we  discover  any  property  in  a  sensible 
quality,  which  at  once  refers  us  to  number  or  space,  we 
readily  take  this  property  as  a  measure ;  and  thus  we 
make  a  transition  from  quality  to  quantity.  Thus  Pto 
lemy  in  the  third  chapter  of  the  First  Book  of  his  Har 
monics  begins  thus :  "  As  to  the  differences  which  exist 
in  sounds  both  in  quality  and  in  quantity,  if  we  consider 
that  difference  which  refers  to  the  acuteness  and  grave- 
ness,  we  cannot  at  once  tell  to  which  of  the  above  two 
classes  it  belongs,  till  we  have  considered  the  causes  of 
such  symptoms."  But  at  the  end  of  the  chapter,  having 
satisfied  himself  that  grave  sounds  result  from  the  mag 
nitude  of  the  string  or  pipe,  other  things  being  equal, 
he  infers,  "  Thus  the  difference  of  acute  and  grave  ap 
pears  to  be  a  difference  of  quantity" 

In  the  same  manner,  in  order  to  form  Secondary 
Mechanical  Sciences  respecting  any  of  the  other  pro 
perties  of  bodies,  we  must  reduce  these  properties  to  a 
dependence  upon  quantity,  and  thus  make  them  subject 
to  measurement.  We  cannot  obtain  any  sciential  truths 
respecting  the  comparison  of  sensible  qualities,  till  we 
have  discovered  measures  and  scales  of  the  qualities 
which  we  have  to  consider;  and  accordingly,  some  of 
the  most  important  steps  in  such  sciences  have  been  the 
establishment  of  such  measures  and  scales,  and  the  inven 
tion  of  the  requisite  instruments. 

The  formation  of  the  mathematical  sciences  which 
VOL.  i.   w.  P.  Y 


rest  upon  the  measures  of  the  intensity  of  sensible  qua 
lities  took  place  mainly  in  the  course  of  the  last  century. 
Perhaps  we  may  consider  Lambert,  a  mathematician 
who  resided  in  Switzerland,  and  published  about  1750, 
as  the  person  who  first  clearly  felt  the  importance  of 
establishing-  such  sciences.  His  Photometry,  Pyrometry, 
HygTometry,  are  examples  of  the  systematic  reduction 
of  sensible  qualities  (light,  heat,  moisture)  to  modes  of 
numerical  measurement. 

We  now  proceed  to  speak  of  such  modes  of  measure 
ment  with  regard  to  the  most  obvious  properties  of 

SECT.  II.— The  Musical  Scale. 

3.  THE  establishment  of  the  Harmonic  Canon,  that 
is,  of  a  Scale  and  Measure  of  the  musical  place  of  notes, 
in  the  relation  of  high  and  low,  was  the  first  step  in  the 
science  of  Harmonics.  The  perception  of  the  differences 
and  relations  of  musical  sounds  is  the  office  of  the  sense 
of  hearing ;  but  these  relations  are  fixed,  and  rendered 
accurately  recognizable  by  artificial  means.  "Indeed, 
in  all  the  senses,"  as  Ptolemy  truly  says  in  the  opening 
of  his  Harmonics,  "  the  sense  discovers  what  is  approxi 
mately  true,  and  receives  accuracy  from  another  quarter: 
the  reason  receives  the  approximately-true  from  another 
quarter,  and  discovers  the  accurate  truth."  We  can 
have  no  measures  of  sensible  qualities  which  do  not 
ultimately  refer  to  the  sense ; — whether  they  do  this 
immediately,  as  when  we  refer  Colours  to  an  assumed 
Standard ;  or  mediately,  as  when  we  measure  Heat  by 
Expansion,  having  previously  found  by  an  appeal  to 
sense  that  the  expansion  increases  with  the  heat.  Such 
relations  of  sensible  qualities  cannot  be  described  in 
words,  and  can  only  be  apprehended  by  their  appropriate 
faculty.  The  faculty  by  which  the  relations  of  sounds 


are  apprehended  is  a  musical  ear  in  the  largest  accep 
tation  of  the  term.  In  this  signification  the  faculty  is 
nearly  universal  among  men;  for  all  persons  have  musical 
ears  sufficiently  delicate  to  understand  and  to  imitate 
the  modulations  corresponding  to  various  emotions  in 
speaking;  which  modulations  depend  upon  the  succes 
sion  of  acuter  and  graver  tones.  These  are  the  relations 
now  spoken  of,  and  these  are  plainly  perceived  by  per 
sons  who  have  very  imperfect  musical  ears,  according  to 
the  common  use  of  the  phrase.  But  the  relations  of 
tones  which  occur  in  speaking  are  somewhat  indefinite ; 
and  in  forming  that  musical  scale  which  is  the  basis  of 
our  science  upon  the  subject,  we  take  the  most  definite 
and  marked  of  such  relations  of  notes ;  such  as  occur, 
not  in  speaking  but  in  singing.  Those  musical  relations 
of  two  sounds  which  we  call  the  octare,  the  fftli,  the 
fourth,  the  third,  are  recognized  after  a  short  familiarity 
with  them.  These  chords  or  intervals  are  perceived  to 
have  each  a  peculiar  character,  which  separates  them 
from  the  relations  of  two  sounds  taken  at  random,  and 
makes  it  easy  to  know  them  when  sung  or  played  on 
an  instrument ;  and  for  most  persons,  not  difficult  to 
sing  the  sounds  in  succession  exactly,  or  nearly  correct. 
These  musical  relations,  or  concords,  then,  are  the  ground 
work  of  our  musical  standard.  But  how  are  we  to  name 
these  indescribable  sensible  characters?  how  to  refer, 
with  unerring  accuracy,  to  a  type  which  exists  only  in 
our  own  perceptions?  We  must  have  for  this  purpose 
a  Scale  and  a  Standard. 

The  Musical  Scale  is  a  series  of  eight  notes,  ascend 
ing  by  certain  steps  from  the  first  or  key-note  to  the 
octave  above  it,  each  of  the  notes  being  fixed  by  such 
distinguishable  musical  relations  as  we  have  spoken  of 
above.  We  may  call  these  notes  c,  n,  E,  F,  G,  A,  n,  c ; 
and  we  may  then  say  that  <;  is  determined  by  its  being  a 

Y  v 


fifth  above  c ;  D  by  its  being  a  fourth  below  G  ;  E  by  its 
being  a  third  above  c ;  and  similarly  of  the  rest.  It 
will  be  recollected  that  the  terms  a  fifth,  a  fourth,  a 
third,  have  hitherto  been  introduced  as  expressing  cer 
tain  simple  and  indescribable  musical  relations  among 
sounds,  which  might  have  been  indicated  by  any  other 
names.  Thus  we  might  call  the  fifth  the  dominant,  and 
the  fourth  the  suldominant,  as  is  done  in  one  part  of 
musical  science.  But  the  names  we  have  used,  which 
are  the  common  ones,  are  in  fact  derived  from  the  num 
ber  of  notes  which  these  intervals  include  in  the  scale 
obtained  in  the  above  manner.  The  notes  c,  D,  E,  F,  G, 
being  five,  the  interval  from  c  to  G  is  a  fifth,  and  so  of 
the  rest.  The  fixation  of  this  scale  gave  the  means  of 
describing  exactly  any  note  which  occurs  in  the  scale, 
and  the  method  is  easily  applicable  to  notes  above  and 
below  this  range ;  for  in  a  series  of  sounds  higher  or 
lower  by  an  octave  than  this  standard  series,  the  ear 
discovers  a  recurrence  of  the  same  relations  so  exact, 
that  a  person  may  sometimes  imagine  he  is  producing 
the  same  notes  as  another  when  he  is  singing  the  same 
air  an  octave  higher.  Hence  the  next  eight  notes  may 
be  conveniently  denoted  by  a  repetition  of  the  same 
letters,  as  the  first ;  thus,  c,  D,  E,  F,  G,  A,  B,  c,  d,  e,  f,  g, 
a,  b ;  and  it  is  easy  to  devise  a  continuation  of  such 
cycles.  And  other  admissible  notes  are  designated  by  a 
further  modification  of  the  standard  ones,  as  by  making 
each  note  flat  or  sharp;  which  modification  it  is  not 
necessary  here  to  consider,  since  our  object  is  only  to 
show  how  a  standard  is  attainable,  and  how  it  serves  the 
ends  of  science. 

We  may  observe,  however,  that  the  above  is  not  an 
exact  account  of  the  first,  or  early  Greek  scale ;  for  this 
scale  was  founded  on  a  primary  division  of  the  interval 
of  two  octaves  (the  extreme  range  which  it  admitted) 


into  five  tetrachords,  each  tetrachord  including  the  in 
terval  of  a  fourth.  All  the  notes  of  this  series  had 
different  names  borrowed  from  this  division* ;  thus  mese 
was  the  middle  or  key-note;  the  note  below  it  was 
lichanos  meson,  the  next  below  was  parypate  meson,  the 
next  lower,  hypate  meson.  The  fifth  above  mese  was 
nete  diazeugmendn,  the  octave  was  nete  hyperbolcedn. 

4.  But  supposing  a  complete  system  of  such  denomi 
nations  established,  how  could  it  be  with  certainty  and 
rigour  applied  ?  The  human  ear  is  fallible,  the  organs 
of  voice  imperfectly  obedient;  if  this  were  not  so,  there 
would  be  no  such  thing  as  a  good  ear  or  a  good  voice. 
What  means  can  be  devised  of  finding  at  will  a  perfect 
concord,  a  fifth  or  a  fourth  ?  Or  supposing  such  con 
cords  fixed  by  an  acknowledged  authority,  how  can  they 
be  referred  to,  and  the  authority  adduced?  How  can 
we  enact  a  Standard  of  sounds  ? 

A  Standard  was  discovered  in  the  Monochord.  A 
musical  string  properly  stretched,  may  be  made  to  pro 
duce  different  notes,  in  proportion  as  we  intercept  a 
longer  or  shorter  portion,  and  make  this  portion  vibrate. 
The  relation  of  the  length  of  the  strings  which  thus 
sound  the  two  notes  G  and  c  is  fixed  and  constant,  and 
the  same  is  true  of  all  other  notes.  Hence  the  musical 
interval  of  any  notes  of  which  we  know  the  places  in 
the  musical  scale,  may  be  reproduced  by  measuring  the 
lengths  of  string  which  are  known  to  give  them.  If  c 
be  of  the  length  180,  D  is  169,  E  is  144,  F  is  135,  G  is 
1 20 ;  and  thus  the  musical  relations  are  reduced  to 
numerical  relations,  and  the  monochord  is  a  complete 
and  perfect  Tonometer. 

We  have  here  taken  the  length  of  the  string  as  the 
measure  of  the  tone :  but  we  may  observe  that  there  is 
in  us  a  necessary  tendency  to  assume  that  the  ground 
*  Burney's  History  of  Music,  Vol.  r.  p.  28. 


of  this  measure  is  to  be  sought  in  some  ulterior  cause ; 
and  when  we  consider  the  matter  further,  we  find  this 
cause  in  the  frequency  of  these  vibrations  of  the  string. 
The  truth  that  the  same  note  must  result  from  the  same 
frequency  of  vibration  is  readily  assented  to  on  a  slight 
suggestion  of  experience.  Thus  Mersenne*,  when  he 
undertakes  to  determine  the  frequency  of  vibrations  of  a 
given  sound,  says  "Supponendum  est  quoscunque  nervos 
et  quaslibet  chordas  unisonum  facientes  eundem  efficere 
numerum  rccursuum  codem  vel  equali  tempore,  quod 
perpetua  constat  experientia."  And  he  proceeds  to 
apply  it  to  cases  where  experience  could  not  verify  this 
assertion,  or  at  least  had  not  verified  it,  as  to  that  of 

The  pursuit  of  these  numerical  relations  of  tones 
forms  the  science  of  Harmonics  ;  of  which  here  we  do 
not  pretend  to  give  an  account,  but  only  to  show,  how 
the  invention  of  a  Scale  and  Nomenclature,  a  Standard 
and  Measure  of  the  tone  of  sounds,  is  its  necessary  basis. 
We  will  therefore  now  proceed  to  speak  of  another  sub 
ject;  colour. 

SECT.  III. — Scales  of  Colour. 

5.  The  Prismatic  Scale  of  Colour. — A  SCALE  of 
Colour  must  depend  originally  upon  differences  discern 
ible  by  the  eye,  as  a  scale  of  notes  depends  on  differences 
perceived  by  the  ear.  In  one  respect  the  difficulty  is 
greater  in  the  case  of  the  visible  qualities,  for  there  are 
no  relations  of  colour  which  the  eye  peculiarly  singles 
out  and  distinguishes,  as  the  ear  selects  and  distinguishes 
an  octave  or  a  fifth.  Hence  we  are  compelled  to  take 
an  arbitrary  scale;  and  we  have  to  find  one  which  is 
fixed,  and  which  includes  a  proper  collection  of  colours. 
The  prismatic  spectrum,  or  coloured  image  produced 

*   Hannonia,  Lib.  11.  Prop.  19. 


when  a  small  beam  of  light  passes  obliquely  through 
uny  transparent  surface  (as  the  surface  of  a  prism  of 
glass,)  otters  an  obvious  Standard  as  far  as  it  is  appli 
cable.  Accordingly  colours  have,  for  various  purposes, 
been  designated  by  their  place  in  the  spectrum  ever 
since  the  time  of  Newton ;  and  we  have  thus  a  means  of 
referring  to  such  colours  as  are  included  in  the  series 
red,  orange,  yellow,  green,  blue,  violet,  indigo,  and  the 
intermediate  tints. 

But  this  scale  is  not  capable  of  numerical  precision. 
If  the  spectrum  could  be  exactly  defined  as  to  its  ex 
tremities,  and  if  these  colours  occupied  always  the  same 
proportional  part  of  it,  we  might  describe  any  colour  in 
the  above  series  by  the  measure  of  its  position.  But 
the  fact  is  otherwise.  The  spectrum  is  too  indefinite  in 
its  boundaries  to  afford  any  distinct  point  from  which 
we  may  commence  our  measures ;  and  moreover  the 
spectra  produced  by  different  transparent  bodies  differ 
from  each  other.  Newton  had  supposed  that  the  spec 
trum  and  its  parts  wrere  the  same,  so  long  as  the  refrac 
tion  was  the  same ;  but  his  successors  discovered  that, 
with  the  same  amount  of  refraction  in  different  kinds  of 
glass,  there  are  different  magnitudes  of  the  spectrum ; 
and  what  is  still  worse  with  reference  to  our  present 
purpose,  that  the  spectra  from  different  glasses  have 
the  colours  distributed  in  different  proportions.  In  order, 
therefore,  to  make  the  spectrum  the  scale  of  colour,  we 
must  assume  some  fixed  substance ;  for  instance,  wre  may 
take  water,  and  thus  a  series  approaching  to  the  colours 
of  the  rainbow  will  be  our  standard.  But  we  should 
still  have  an  extreme  difficulty  in  applying  such  a  rule. 
The  distinctions  of  colour  which  the  terms  of  common 
language  express,  are  not  used  with  perfect  unanimity 
or  with  rigorous  precision.  What  one  person  calls  bluish 
green  another  calls  greenish  blue.  Nobody  can  say 


what  is  the  precise  boundary  between  red  and  orange. 
Thus  the  prismatic  scale  of  colour  was  incapable  of 
mathematical  exactness,  and  this  inconvenience  was  felt 
up  to  our  own  times. 

But  this  difficulty  was  removed  by  a  curious  dis 
covery  of  Wollaston  and  Fraunhofer ;  who  found  that 
there  are,  in  the  solar  spectrum,  certain  fine  black  Lines 
which  occupy  a  definite  place  in  the  series  of  colours, 
and  can  be  observed  with  perfect  precision.  We  have 
now  no  uncertainty  as  to  what  coloured  light  we  are 
speaking  of,  when  we  describe  it  as  that  part  of  the 
spectrum  in  which  Fraunhofer's  Line  c  or  D  occurs. 
And  thus,  by  this  discovery,  the  prismatic  spectrum  of 
sunlight  became,  for  certain  purposes,  an  exact  Chroma- 

6.  Neifton's  Scale  of  Colours. — Still,  such  a  standard, 
though  definite,  is  arbitrary  and  seemingly  anomalous. 
The  lines  A,  B,  c,  D,  &c.,  of  Fraunhofer's  spectrum  are 
distributed  without  any  apparent  order  or  law ;  and  we 
do  not,  in  this  way,  obtain  numerical  measures,  which  is 
what,  in  all  cases,  we  desire  to  have.  Another  discovery 
of  Newton,  however,  gives  us  a  spectrum  containing  the 
same  colours  as  the  prismatic  spectrum,  but  produced  in 
another  way,  so  that  the  colours  have  a  numerical  rela 
tion.  I  speak  of  the  laws  of  the  colours  of  thin  plates. 
The  little  rainbows  which  we  sometimes  see  in  the  cracks 
of  broken  glass  are  governed  by  fixed  and  simple  laws. 
The  kind  of  colour  produced  at  any  point  depends  on 
the  thickness  of  the  thin  plate  of  air  included  in  the  fis 
sure.  If  the  thickness  be  eight-millionths  of  an  inch, 
the  colour  is  orange,  if  fifteen-millionths  of  an  inch,  we 
have  green,  and  so  on  ;  and  thus  these  numbers  which 
succeed  each  other  in  a  regular  order  from  red  to  indigo, 
give  a  numerical  measure  of  each  colour ;  which  mea 
sure,  when  we  pursue  the  subject,  we  find  is  one  of  the 


bases  of  all  optical  theory.  The  series  of  colours  ob 
tained  from  plates  of  air  of  gradually  increasing  thick 
ness  is  called  Newton  s  Scale  of  Colours ;  but  we  may 
observe  that  this  is  not  precisely  what  we  are  here  speak 
ing  of,  a  scale  of  simple  colours ;  it  is  a  series  produced 
by  certain  combinations,  resulting  from  the  repetition  of 
the  first  spectrum,  and  is  mainly  useful  as  a  standard  for 
similar  phenomena,  and  not  for  colour  in  general.  The 
real  scale  of  colour  is  to  be  found,  as  we  have  said,  in 
the  numbers  which  express  the  thickness  of  the  pro 
ducing  film; — in  the  length  of  a  Jit  in  Newton's  phrase 
ology,  or  the  length  of  an  undulation  in  the  modern 

7.  Scales  of  Impure  Colours. — The  standards  just 
spoken  of  include  (mainly  at  least)  only  pure  and  simple 
colours ;  and  however  complete  they  may  be  for  certain 
objects  of  the  science  of  optics,  they  are  insufficient  for 
other  purposes.  They  do  not  enable  us  to  put  in  their 
place  mixed  and  impure  colours.  And  there  is,  in  the 
case  of  colour,  a  difficulty  already  noticed,  which  does 
not  occur  in  the  case  of  sound  ;  two  notes,  when  sounded 
together,  are  not  necessarily  heard  as  one;  they  are 
recognized  as  still  two,  and  as  forming  a  concord  or  a 
discord.  But  two  colours  form  a  single  colour ;  and  the 
eye  cannot,  in  any  way,  distinguish  between  a  green 
compounded  of  blue  and  yellow,  and  the  simple,  unde- 
composable  green  of  the  spectrum.  By  composition  of 
three  or  more  colours,  innumerable  new  colours  may  be 
generated  which  form  no  part  of  the  prismatic  series ; 
and  by  such  compositions  is  w.oven  the  infinitely  varied 
web  of  colour  which  forms  the  clothing  of  nature.  How 
are  we  to  classify  and  arrange  all  the  possible  colours 
of  objects,  so  that  each  shall  have  a  place  and  name? 
How  shall  we  find  a  chromato  meter  for  impure  as  well 
as  for  pure  colour  ? 


Though  no  optical  investigations  have  depended  on  a 
scale  of  impure  colours,  such  a  scale  has  been  wanted 
and  invented  for  other  purposes ;  for  instance,  in  order 
to  identity  and  describe  objects  of  natural  history.  Not 
to  speak  of  earlier  essays,  we  may  notice  Werner's  No 
menclature  of  Colours,  devised  for  the  purpose  of  de 
scribing  minerals.  This  scale  of  colour  was  far  superior 
to  any  which  had  previously  been  promulgated.  It  was, 
indeed,  arbitrary  in  the  selection  of  its  degrees,  and  in 
a  great  measure  in  their  arrangement ;  and  the  colours 
were  described  by  the  usual  terms,  though  generally 
with  some  added  distinction ;  as  blackish  green,  bluish 
green,  apple-green,  emerald-green.  But  the  great  merit 
of  the  scale  was  its  giving  a  fixed  conventional  meaning 
to  these  terms,  so  that  they  lost  much  of  their  usual 
vagueness.  Thus  apple-green  did  not  mean  the  colour 
of  any  green  apple  casually  taken  ;  but  a  certain  definite 
colour  which  the  student  was  to  bear  in  mind,  whether  or 
not  he  had  ever  seen  an  apple  of  that  exact  hue.  The 
words  were  not  a  description,  but  a  record  of  the  colour  : 
the  memory  was  to  retain  a  sensation,  not  a  name. 

The  imperfection  of  the  system  (arising  from  its  ar 
bitrary  form)  was  its  incompleteness :  however  well  it 
served  for  the  reference  of  the  colours  which  it  did  con 
tain,  it  was  applicable  to  no  others;  and  thus,  though 
Werner's  enumeration  extended  to  more  than  a  hundred 
colours,  there  occur  in  nature  a  still  greater  number 
which  cannot  be  exactly  described  by  means  of  it. 

In  such  cases  the  imclassed  colour  is,  by  the  Werne- 
rians,  defined  by  stating  it  as  intermediate  between  two 
others :  thus  we  have  an  object  described  as  between 
emerald-green  and  grass-green.  The  eye  is  capable  of 
perceiving  a  gradation  from  one  colour  to  another  ;  such 
as  may  be  produced  by  a  gradual  mixture  in  various 
ways.  And  if  we  image  to  ourselves  such  a  mixture,  we 


can  compare  with  it  a  given  colour.  But  in  employing 
this  method  we  have  nothing  to  tell  us  in  what  part  of 
the  scale  we  must  seek  for  an  approximation  to  our  un- 
classed  colour.  We  have  no  rule  for  discovering  where 
we  are  to  look  for  the  boundaries  of  the  definition  of  a 
colour  which  the  Wernerian  series  does  not  supply. 
For  it  is  not  always  between  contiguous  members  of  the 
series  that  the  undescribed  colour  is  found.  If  we  place 
emerald-green  between  apple-green  and  grass-green,  we 
may  yet  have  a  colour  intermediate  between  emerald- 
green  and  leek-green ;  and,  in  fact,  the  Wernerian  series 
of  colours  is  destitute  of  a  principle  of  self-arrangement 
and  gradation ;  and  is  thus  necessarily  and  incurably 

8.  We  should  have  a  complete  Scale  of  Colours,  if 
we  could  form  a  series  including  all  colours,  and  arranged 
so  that  each  colour  was  intermediate  in  its  tint  between 
the  adjacent  terms  of  the  series ;  for  then,  whether  we 
took  many  or  few  of  the  steps  of  the  series  for  our 
standard  terms,  the  rest  could  be  supplied  by  the  law  of 
continuity;  and  any  given  colour  would  either  cor 
respond  to  one  of  the  steps  of  our  scale  or  fall  between 
two  intermediate  ones.  The  invention  of  a  Chroma- 
,  tometer  for  Impure  Colours,  therefore,  requires  that  we 
should  be  able  to  form  all  possible  colours  by  such  inter 
mediation  in  a  systematic  manner ;  that  is,  by  the  mix 
ture  or  combination  of  certain  elementary  colours  ac 
cording  to  a  simple  rule :  and  we  are  led  to  ask  whether 
such  a  process  has  been  shown  to  be  possible. 

The  colours  of  the  prismatic  spectrum  obviously  do 
form  a  continuous  series ;  green  is  intermediate  between 
its  neighbours  yellow  and  blue,  orange  between  red  and 
yellow ;  and  if  we  suppose  the  two  ends  of  the  spectrum 
bent  round  to  meet  each  other,  so  that  the  arrangement 
of  the  colours  may  be  circular,  the  violet  and  indigo  will 


find  their  appropriate  place  between  the  blue  and  red. 
And  all  the  interjacent  tints  of  the  spectrum,  as  well  as 
the  ones  thus  named,  will  result  from  such  an  arrange 
ment.     Thus  all  the  pure  colours  are  produced  by  com 
binations  two  and  two    of  three  primary  colours,  red, 
yellow,    and    blue ;    and    the    question    suggests   itself 
whether  these  three  are   not   really  the   only  primary 
colours,  and  whether  all  the  impure  colours  do  not  arise 
from    mixtures    of  the   three   in    various    proportions. 
There  are  various  modes  in  which  this  suggestion  may 
be  applied  to  the  construction  of  a  scale  of  colours ;  but 
the  simplest,  and  the  one  which  appears  really  to  verify 
the  conjecture  that  all  possible  colours  may  be  so  ex 
hibited,  is  the  following.     A  certain  combination  of  red, 
yellow,  and  blue,  will  produce  black,  or  pure  grey,  and 
when  diluted,  will  give  all  the  shades  of  grey  which 
intervene  between  black  and  white.     By  adding  various 
shades  of  grey,  then,  to  pure  colours,  we  may  obtain  all 
the  possible  ternary  combinations  of  red,  yellow,   and 
blue  ;  and  in  this  way  it  is  found  that  we  exhaust  the 
range  of  colours.     Thus  the  circle   of  pure  colours  of 
which  we  have  spoken  may  be  accompanied  by  several 
other  circles,  in  which  these  colours  are  tinged  with  a 
less  or  greater  shade  of  grey ;  and  in  this  manner  it  is 
found  that  we  have  a  perfect    chromatometer ;    every 
possible    colour   being   exhibited   either  exactly  or  by 
means  of  approximate  and  contiguous  limits.     The  ar 
rangement  of  colours  has  been  brought  into  this  final 
and  complete  form   by  M.  Merimee,  whose  Chromatic 
Scale  is  published  by  M.  Mirbel  in  his  Elements  of  Bo 
tany.     We  may  observe  that  such  a  standard  affords  us 
a  numerical  exponent  for  every  colour  by  means  of  the 
proportions  of  the  three  primary  colours  which  compose 
it ;  or,  expressing  the  same  result  otherwise,  by  means 
of  the  pure  colour  which  is  involved,  and  the  proportion 


of  grey  by  which  it  is  rendered  impure.  In  such  a 
scale  the  fundamental  elements  would  be  the  precise 
tints  of  red,  yellow,  and  blue  which  are  found  or  as 
sumed  to  be  primary ;  the  numerical  exponents  of  each 
colour  would  depend  upon  the  arbitrary  number  of  de 
grees  which  we  interpose  between  each  two  primary 
colours;  and  between  each  pure  colour  and  absolute 
blackness.  No  such  numerical  scale  has,  however,  as  yet, 
obtained  general  acceptation'-. 

SECT.  IV. — Scales  of  Light. 

9.  Photometer. — ANOTHER  instrument  much  needed 
in  optical  researches  is  a  Photometer,  a  measure  of  the 
intensity  of  light.  In  this  case,  also,  the  organ  of  sense, 
the  eye,  is  the  ultimate  judge ;  nor  has  any  effect  of 
light,  as  light,  yet  been  discovered  which  we  can  sub 
stitute  for  such  a  judgment.  All  instruments,  such  as 
that  of  Leslie,  which  employ  the  heating  effect  of  light, 
or  at  least  all  that  have  hitherto  been  proposed,  are  in 
admissible  as  photometers.  But  though  the  eye  can 

*  The  reference  to  Fraunhofer's  Lines,  as  a  means  of  determining 
the  place  of  a  colour  in  the  prismatic  series,  has  been  objected  to, 
because,  as  is  asserted,  the  colours  which  are  in  the  neighbourhood  of 
each  line  vary  with  the  position  of  the  sun,  state  of  the  atmosphere 
and  the  like.  It  is  very  evident  that  coloured  light  refracted  by  the 
prism  will  not  give  the  same  spectrum  as  white  light.  The  spectrum 
given  by  white  light  is  of  course  the  one  here  meant.  It  is  an  usual 
practice  of  optical  experimenters  to  refer  to  the  colours  of  such  a 
spectrum,  defining  them  by  Fraunhofer's  Lines. 

I  do  not  know  whether  it  needs  explanation  that  the  "  first  spec 
trum"  in  Newton's  rings  is  a  ring  of  the  prismatic  colours. 

I  have  not  had  an  opportunity  of  consulting  Lambert's  Photometria, 
sive  de  mensura  et  gradibus  luminis,  colorum,  el  umbra;,  published  in 
17<>0,  nor  Mayer's  Commentatio  de  Affinitate  Colorum,  (1758,)  in 
which,  I  believe,  he  describes  a  chromatometer.  The  present  work  is 
not  intended  to  be  complete  as  a  history ;  and  I  hope  I  have  given 
sufficient  historical  detail  to  answer  its  philosophical  purpose. 


judge  of  two  surfaces  illuminated  by  light  of  the  same 
colour,  and  can  determine  when  they  are  equally  bright, 
or  which  is  the  brighter,  the  eye  can  by  no  means  decide 
at  sight  the  proportion  of  illumination.  How  much  in 
such  judgments  we  are  affected  by  contrast,  is  easily  seen 
when  we  consider  how  different  is  the  apparent  bright 
ness  of  the  moon  at  mid-day  and  at  midnight,  though 
the  light  which  we  receive  from  her  is,  in  fact,  the  same 
at  both  periods.  In  order  to  apply  a  scale  in  this  case, 
we  must  take  advantage  of  the  known  numerical  rela 
tions  of  lidit.  We  are  certain  that  if  all  other  illumi- 


nation  be  excluded,  two  equal  luminaries,  under  the 
same  circumstances,  will  produce  an  illumination  twice 
as  great  as  one  does ;  and  we  can  easily  prove,  from  ma 
thematical  considerations,  that  if  light  be  not  enfeebled 
by  the  medium  through  which  it  passes,  the  illumination 
on  a  given  surface  will  diminish  as  the  square  of  the 
distance  of  the  luminary  increases.  If,  therefore,  we 
can  by  taking  a  fraction  thus  known  of  the  illuminating 
effect  of  one  luminary,  make  it  equal  to  the  total  effect 
of  another,  of  which  equality  the  eye  is  a  competent 
judge,  we  compare  the  effects  of  the  two  luminaries.  In 
order  to  make  this  comparison  we  may,  with  Rumford, 
look  at  the  shadows  of  the  same  object  made  by  the  two 
lights,  or  with  Ritchie,  we  may  view  the  brightness  pro 
duced  on  two  contiguous  surfaces,  framing  an  apparatus 
so  that  the  equality  may  be  brought  about  by  proper 
adjustment ;  and  thus  a  measure  will  become  practica 
ble.  Or  we  may  employ  other  methods  as  was  done  by 
Wollaston "-,  who  reduced  the  light  of  the  sun  by  observ 
ing  it  as  reflected  from  a  bright  globule,  and  thus  found 
the  light  of  the  sun  to  be  10,000,000,000  times  that  of 
Sirius,  the  brightest  fixed  star.  All  these  methods  are 
inaccurate,  even  as  methods  of  comparison ;  and  do  not 

*    Phil.   Trans.,  1829,  p.  19. 


offer  any  fixed  or  convenient  numerical  standard ;  but 
none  better  have  yet  been  devised"". 

10.  Cyanometer. — As  we  thus  measure  the  brightness 
of  a  colourless  light,  we  may  measure  the  intensity  of 
any  particular  colour  in  the  same  way;  that  is,  by  apply 
ing  a  standard  exhibiting  the  gradations  of  the  colour  in 
question  till  we  find  a  shade  which  is  seen  to  agree  with 
the  proposed  object.     Such  an  instrument  we  have  in 
the  Cyanometer,  which  was  invented  by  Saussure  for  the 
purpose  of  measuring  the  intensity  of  the  blue  colour  of 
the  sky.     We  may  introduce  into  such  an  instrument  a 
numerical  scale,  but  the  numbers  in  such  a  scale  will  be 
altogether  arbitrary. 

SECT.  V. — Scales  of  Heat. 

11.  Thermometers. — WHEN  we  proceed  to  the  sensa 
tion  of  heat,  and  seek  a  measure  of  that  quality,  we  find, 
at  first  sight,  new  difficulties.     Our  sensations  of  this 
kind  are  more  fluctuating  than  those  of  vision ;  for  we 
know  that  the  same  object  may  feel  warm  to  one  hand 
and  cold  to  another  at  the  same  instant,  if  the  hands 
have  been  previously  cooled  and  warmed  respectively. 
Nor  can  we  obtain  here,  as  in  the  case  of  light,  self-evi 
dent  numerical  relations  of  the  heat  communicated   in 
given  circumstances ;  for  we  know  that  the  effect  so  pro 
duced  will  depend  on   the  warmth  of  the  body  to  be 
heated,  as  well  as  on  that  of  the  source  of  heat ;  the 
summer  sun,  which  warms  our  bodies,  will  not  augment 
the  heat  of  a  red-hot  iron.      The  cause  of  the  differ 
ence  of  these  cases  is,  that  bodies  do  not  receive  the 
whole  of  their  heat,  as  they  receive  the  whole  of  their 
light,  from  the  immediate  influence  of  obvious  external 

*  Improved  Photometers  have  been  devised  by  Professor  Wheat- 
stone,  Professor  Potter,  and  Professor  Steinheil ;  but  they  depend  upon 
principles  similar  to  those  mentioned  in  the  text. 


agents.  There  is  no  readily-discovered  absolute  cold, 
corresponding  to  the  absolute  darkness  which  we  can 
easily  produce  or  imagine.  Hence  we  should  be  greatly 
at  a  loss  to  devise  a  Thermometer,  if  we  did  not  find  an 
indirect  effect  of  heat  sufficiently  constant  and  measurable 
to  answer  this  purpose.  We  discover,  however,  such  an 
effect  in  the  expansion  of  bodies  by  the  effect  of  heat. 

12.  Many  obvious  phenomena  show  that  air,  under 
given  circumstances,  expands  by  the  effect  of  heat ;  the 
same  is  seen  to  be  true  of  liquids,  as  of  water,  and  spirit 
of  wine ;  and  the  property  is  found  to  belong  also  to  the 
metallic  fluid,  quicksilver.     A  more  careful  examination 
showed  that  the  increase  of  bulk  in  some  of  these  bodies 
by  increase  of  heat  was  a  fact  of  a  nature  sufficiently 
constant  and  regular  to  afford  a  means  of  measuring  that 
previously  intangible  quality ;  and  the  Thermometer  was 
invented.     There   were,  however,   many    difficulties   to 
overcome,  and  many  points  to  settle,  before  this  instru 
ment  was  fit  for  the  purposes  of  science. 

An  explanation  of  the  way  in  which  this  was  done 
necessarily  includes  an  important  chapter  of  the  history 
of  Thermotics.  We  must  now,  therefore,  briefly  notice 
historically  the  progress  of  the  Thermometer.  The  lead 
ing  steps  of  this  progress,  after  the  first  invention  of  the 
instrument,  were — The  establishment  of  fixed  points  in 
the  thermometric  scale — The  comparison  of  the  scales 
of  different  substances — And  the  reconcilement  of  these 
differences  by  some  method  of  interpreting  them  as  indi 
cations  of  the  absolute  quantity  of  heat. 

13.  It  would  occupy  too  much  space  to  give  in  detail 
the  history  of  the  successive  attempts  by  which  these 
steps  were  effected.     A  thermometer   is  described  by 
Bacon  under  the  title  Vitrum  Calendare ;  this  was  an 
air  thermometer.   Newton  used  a  thermometer  of  linseed 
oil,  and  he  perceived  that  the  first  step  requisite  to  give 


value  to  such  an  instrument  was  to  fix  its  scale ;  accord 
ingly  he  proposed  his  Scala  Graduum  Caloris*.  But 
when  thermometers  of  different  liquids  were  compared, 
it  appeared,  from  their  discrepancies,  that  this  fixation 
of  the  scale  of  heat  was  more  difficult  than  had  been 
supposed.  It  was,  however,  effected.  Newton  had  taken 
freezing  water,  or  rather  thawing  snow,  as  the  zero  of 
his  scale,  which  is  really  a  fixed  point;  Halley  and  Amon- 
tons  discovered  (in  1693  and  1702)  that  the  heat  of 
boiling  water  is  another  fixed  point ;  and  Daniel  Gabriel 
Fahrenheit,  of  Dantzig,  by  carefully  applying  these  two 
standard  points,  produced,  about  1714,  thermometers, 
which  were  constantly  consistent  with  each  other.  This 
result  was  much  admired  at  the  time,  and  was,  in  fact, 
the  solution  of  the  problem  just  stated,  the  fixation  of 
the  scale  of  heat. 

14.  But  the  scale  thus  obtained  is  a  conventional 
not  a  natural  scale.    It  depends  upon  the  fluid  employed 
for  the  thermometer.     The  progress  of  expansion  from 
the  heat  of  freezing  to  that  of  boiling  water  is  different 
for  mercury,  oil,  water,  spirit  of  wine,  air.     A  degree  of 
heat   which   is    half-way    between   these  two   standard 
points  according  to  a  mercurial  thermometer,  will  be  . 
below  the  half-way  point  in  a  spirit  thermometer,  and 
above  it  in  an  air  thermometer.     Each  liquid  has  its 
own  march  in  the  course  of  its  expansion.     Deluc  and 
others   compared  the  marches  of  various  liquids,  and 
thus  made  what  we  may  call  a  concordance  of  thermo 
meters  of  various  kinds. 

15.  Here  the  question  further  occurs :  Is  there  not 
some  natural  measure  of  the  degrees  of  heat  ?     It  ap 
pears  certain  that  there  must  be  such  a  measure,  and 
that  by  means  of  it  all  the  scales  of  different   liquids 
must  be  reconciled.     Yet  this  does  not  seem  to  have 

*    Phil.  Trans.,  1701. 
VOL.  I.     W.  P.  / 


occurred  at  once  to  men's  minds.  Deluc,  in  speaking 
of  the  researches  which  we  have  just  mentioned,  says*, 
"When  I  undertook  these  experiments,  it  never  once 
came  into  my  thoughts  that  they  could  conduct  me  with 
any  probability  to  a  table  of  real  degrees  of  heat.  But 
hope  grows  with  success,  and  desire  with  hope."  Accord 
ingly  he  pursued  this  inquiry  for  a  long  course  of  years. 

What  are  the  principles  by  which  we  are  to  be 
guided  to  the  true  measure  of  heat  ?  Here,  as  in  all  the 
sciences  of  this  class,  we  have  the  general  principle,  that 
the  secondary  quality,  heat,  must  be  supposed  to  be  per 
ceived  in  some  way  by  a  material  medium  or  fluid.  If 
we  take  that  which  is,  perhaps,  the  simplest  form  of  this 
hypothesis,  that  the  heat  depends  upon  the  quantity  of 
this  fluid,  or  caloric,  which  is  present,  we  shall  find  that 
we  are  led  to  propositions  which  may  serve  as  a  foun 
dation  for  a  natural  measure  of  heat.  The  Method  of 
Mixtures  is  one  example  of  such  a  result.  If  we  mix 
together  two  pints  of  water,  one  hot  and  one  cold,  is  it 
not  manifest  that  the  temperature  of  the  mixture  must 
be  midway  between  the  two?  Each  of  the  two  portions 
brings  with  it  its  own  heat.  The  whole  heat,  or  caloric, 
of  the  mixture  is  the  sum  of  the  two ;  and  the  heat  of 
each  half  must  be  the  half  of  this  sum,  and  therefore  its 
temperature  must  be  intermediate  between  the  tempe 
ratures  of  the  equal  portions  which  were  mixed.  Deluc 
made  experiments  founded  upon  this  principle,  and  was 
led  by  them  to  conclude  that  "the  dilatations  of  mer 
cury  follow  an  accelerated  march  for  successive  equal 
augmentations  of  heat." 

But  there  are  various  circumstances  which  prevent 
this  method  of  mixtures  from  being  so  satisfactory  as 
at  first  sight  it  seems  to  promise  to  be.  The  different 
capacities  for  heat  of  different  substances,  and  even  of 

*  Modif.  tie  1'Ahnosph.,  1782,  p.  303. 


the  same  substance  at  different  temperatures,  introduce 
much  difficulty  into  the  experiments;  and  this  path  of 
inquiry  has  not  yet  led  to  a  satisfactory  result. 

16.  Another  mode   of  inquiring  into   the  natural 
measure  of  heat  is  to  seek  it  by  researches  on  the  laic 
of  cooling  of  hot  bodies.     If  we  assume  that  the  process 
of  cooling  of  hot  bodies  consists  in  a  certain  material 
heat  flying  off,  we  may,  by  means  of  certain  probable 
hypotheses,  determine  mathematically  the  law  according 
to  which  the  temperature  decreases  as  time  goes  on ;  and 
we  may  assume  that  to  be  the  true  measure  of  tempe 
rature  which  gives  to  the  experimental  law  of  cooling 
the  most  simple  and  probable  form. 

It  appears  evident  from  the  most  obvious  conceptions 
which  we  can  form  of  the  manner  in  which  a  body  parts 
with  its  superabundant  heat,  that  the  hotter  a  body  is, 
the  faster  it  cools ;  though  it  is  not  clear  without  expe 
riment,  by  what  law  the  rate  of  cooling  will  depend  upon 
the  heat  of  the  body.  Newton  took  for  granted  the 
most  simple  and  seemingly  natural  law  of  this  depend 
ence  :  he  supposed  the  rate  of  cooling  to  be  proportional 
to  the  temperature,  and  from  this  supposition  he  could 
deduce  the  temperature  of  a  hot  iron,  calculating  from 
the  original  temperature  and  the  time  during  which  it 
had  been  cooling.  By  calculation  founded  on  such  a 
basis,  he  graduated  his  thermometer. 

17.  But  a  little  further  consideration  showed 
the  rate  of  cooling  of  hot  bodies  depended  upon  the 
temperature  of  the  surrounding  bodies,  as  well  as  upon 
its  own  temperature.     Prevost's  Theory  of  Exchanges* 
was  propounded  with  a  view  of  explaining  this  depend 
ence,  and  was  generally  accepted.     According  to  this 
theory,  all  bodies  radiate  heat  to  one  another,  and  are 
thus  constantly  giving  and  receiving  heat;  and  a  body 

*  Pecherches  stir  In  Cha/cur,  1791-    Hi»t.  Ind.  Sci.,  B.  x.  c.  i.  sect.  2. 


which  is  hotter  than  surrounding  bodies,  cools  itself, 
and  warms  the  surrounding  bodies,  by  an  exchange  of 
heat  for  heat,  in  which  they  are  the  gainers.  Hence  if 
6  be  the  temperature  of  the  bodies,  or  of  the  space,  by 
which  the  hot  body  is  surrounded,  and  0  +  t  the  tempe 
rature  of  the  hot  body,  the  rate  of  cooling  will  depend 
upon  the  excess  of  the  radiation  for  a  temperature  6>  +  t, 
above  the  radiation  for  a  temperature  0. 

Accordingly,  in  the  admirable  researches  of  MM. 
Dulong  and  Petit  upon  the  cooling  of  bodies,  it  was 
assumed  that  the  rate  of  cooling  of  the  hot  body  was 
represented  by  the  excess  of  F  (0  +  t)  above  F  (0);  where 
F  represented  some  mathematical  function,  that  is,  some 
expression  obtained  by  arithmetical  operations  from  the 
temperatures  9  +  t  and  0:  although  what  these  operations 
are  to  be,  was  left  undecided,  and  was  in  fact  determined 
by  the  experiments.  And  the  result  of  their  investiga 
tions  was,  that  the  function  is  of  this  kind : — when  -the 
temperature  increases  by  equal  intervals,  the  function 
increases  in  a  continued  geometric  proportion""".  This 
was,  in  fact,  the  same  law  which  had  been  assumed  by 
Newton  and  others,  with  this  difference,  that  they  had 
neglected  the  term  which  depends  upon  the  temperature 
of  the  surrounding  space. 

18.  This  law  falls  in  so  well  with  the  best  concep 
tions  we  can  form  of  the  mechanism  of  cooling  upon  the 
supposition  of  a  radiant  fluid  caloric,  that  it  gives  great 
probability  to  the  scale  of  temperature  on  which  the 
simplicity  of  the  result  depends.  Now  the  temperatures 
in  the  formula?  just  referred  to  were  expressed  by  means 
of  the  air  thermometer.  Hence  MM.  Dulong  and  Petit 
justly  state  that  while  all  different  substances  employed 

"  The  formula  for  the  rate  of  cooling  is  mae+t  —  ma0.  where  the 
quantity  m  depends  upon  the  nature  of  the  body,  the  state  of  its  sur 
face,  and  other  circumstances. — Ann.  Chim.  vn.  loO. 


as  thermometers  give  different  laws  of  thermotical  phe 
nomena,  their  own  success  in  obtaining  simple  and 
general  laws  by  means  of  the  air  thermometer,  is  a  strong 
recommendation  of  that  as  the  natural  scale  of  heat. 
They  add*,  "  The  well-known  uniformity  of  the  principal 
physical  properties  of  all  gases,  and  especially  the  per 
fect  identity  of  their  laws  of  dilatation  by  heat,  [a  very 
important  discovery  of  Dalton  and  Gay  Lussacf,]  make 
it  very  probable  that  in  this  class  of  bodies  the  disturb 
ing  causes  have  not  the  same  influence  as  in  solids  and 
liquids ;  and  consequently  that  the  changes  of  bulk  pro 
duced  by  the  action  of  heat  are  here  in  a  more  imme 
diate  dependence  on  the  force  which  produces  them." 

19.  Still  we  cannot  consider  this  point  as  settled 
till  we  obtain  a  more  complete  theoretical  insight  into 
the  nature  of  heat  itself.  If  it  be  true  that  heat  con 
sists  in  the  vibrations  of  a  fluid,  then,  although,  as 
Ampere  has  shown |,  the  laws  of  radiation  will,  on 
mathematical  grounds,  be  the  same  as  they  are  on  the 
hypothesis  of  emission,  we  cannot  consider  the  natural 
scale  of  heat  as  determined,  till  we  have  discovered  some 
means  of  measuring  the  caloriferous  vibrations  as  we 
measure  luminiferous  vibrations.  We  shall  only  know 
what  the  quantity  of  heat  is  when  we  know  what  heat 
itself  is ; — when  we  have  obtained  a  theory  which  satis 
factorily  explains  the  manner  in  which  the  substance  or 
medium  of  heat  produces  it  effects.  When  we  see  how 
radiation  and  conduction,  dilatation  and  liquefaction,  are 
all  produced  by  mechanical  changes  of  the  same  fluid, 
we  shall  then  see  what  the  nature  of  that  change  is 
which  dilatation  really  measures,  and  what  relation  it 
bears  to  any  more  proper  standard  of  heat. 

We  may  add,  that  while  our  thermotical  theory  is 

*  Annalcsde  Chimie,  vn.  lf>3.  t  Hist.  Ind.  Sci.,  B.  x.  c.  ii.  sect.  1. 

I  //>..  c.  iv. 


still  so  imperfect  as  it  is,  all  attempts  to  divine  the  true 
nature  of  the  relation  between  light  and  heat  are  pre 
mature,  and  must  be  in  the  highest  degree  insecure 
and  visionary.  Speculations  in  which,  from  the  general 
assumption  of  a  caloriferous  and  luminiferous  medium, 
and  from  a  few  facts  arbitrarily  selected  and  loosely 
analyzed,  a  general  theory  of  light  and  heat  is  asserted, 
are  entirely  foreign  to  the  course  of  inductive  science, 
and  cannot  lead  to  any  stable  and  substantial  truth. 

•JO.  Other  Instruments  for  measuring  Heat. — It 
does  not  belong  to  our  present  purpose  to  speak  of 
instruments  of  which  the  object  is  to  measure,  not  sen 
sible  qualities,  but  some  effect  or  modification  of  the 
cause  by  which  such  qualities  are  produced :  such,  for 
instance,  are  the  Calorimeter,  employed  by  Lavoisier 
and  Laplace,  in  order  to  compare  the  specific  lieat  of 
different  substances ;  and  the  Actiiwmeter,  invented  by 
Sir  John  Herschel,  in  order  to  determine  the  effect  of 
the  suns  rai/s  by  means  of  the  heat  which  they  commu 
nicate  in  a  given  time ;  which  effect  is,  as  may  readily 
be  supposed,  very  different  under  different  circumstances 
of  atmosphere  and  position.  The  laws  of  such  effects 
may  be  valuable  contributions  to  our  knowledge  of  heat, 
but  the  interpretation  of  them  must  depend  on  a  pre 
vious  knowledge  of  the  relations  which  temperature  bears 
to  heat,  according  to  the  views  just  explained. 

SECT.  VI. — Scales  of  other  Qualities. 

21.  BEFORE  quitting  the  subject  of  the  measures  of 
sensible  qualities,  we  may  observe  that  there  are  several 
other  such  qualities  for  which  it  would  be  necessary  to 
have  scales  and  means  of  measuring,  in  order  to  make 
any  approach  to  science  on  such  subjects.  This  is  true, 
for  instance,  of  tastes  and  smells.  Indeed  some  attempts 
have  been  made  towards  a  classification  of  the  tastes  of 


sapid  substances,  but  these  have  not  yet  assumed  any 
satisfactory  or  systematic  character ;  and  I  am  not  aware 
that  any  instruments  has  been  suggested  for  measuring 
either  the  flavour  or  the  odour  of  bodies  which  possess 
such  qualities. 

22.  Quality  of  Sounds. — The  same  is  true  of  that 
kind  of  difference  in  sounds  which  is  peculiarly  termed 
their  quality ;  that  character  by  which,  for  instance,  the 
sound  of  a  flute  differs  from  that  of  a  hautbois,  when  the 
note  is  the  same ;  or  a  woman's  voice  from  a  boy's. 

23.  Articulate   Sounds. — There  is   also  in   sounds 
another  difference,  of  which  the  nature  is  still  obscure, 
but  in  reducing  which  to  rule,  and  consequently  to  mea 
sure,    some   progress  has   nevertheless   been  made.     I 
speak  of  the  differences  of  sound  considered  as  articulate. 
Classifications  of  the  sounds  of  the  usual  alphabets  have 
been  frequently  proposed ;  for  instance,  that  which  ar 
ranges  the  consonants  in  the  following  groups  : 

Sharp.  Flat.  Sharp  Aspirate1.  Flat  Aspirate.  Nasal. 

p                  b                      ph  (  /)                  bh  (i>)  m 

k                 g  (hard)          kh                         gh  ng 

t                  (\                       th  (sharp)            th  (flat)  n 

s  x  sh  zh 

It  is  easily  perceived  that  the  relations  of  the  sounds  in 
each  of  these  horizontal  lines  are  analogous ;  and  accord 
ingly  the  rules  of  derivation  and  modification  of  words 
in  several  languages  proceed  upon  such  analogies.  In 
the  same  manner  the  vowels  may  be  arranged  in  an  order 
depending  on  their  sound.  But  to  make  such  arrange 
ments  fixed  and  indisputable,  we  ought  to  know  the 
mechanism  by  which  such  modifications  are  caused.  In 
struments  have  been  invented  by  which  some  of  these 
sounds  can  be  imitated ;  and  if  such  instruments  could 
be  made  to  produce  the  above  series  of  articulate  sounds, 
by  connected  and  regular  processes,  we  should  find,  in 


the  process,  a  measure  of  the  sound  produced.  This 
has  been  in  a  great  degree  effected  for  the  Vowels  by 
Professor  Willis's  artificial  mode  of  imitating  them.  For 
he  finds  that  if  a  musical  reed  be  made  to  sound  through 
a  cylindrical  pipe,  we  obtain  by  gradually  lengthening 
the  cylindrical  pipe,  the  series  of  vowels  I,  E,  A,  o,  u, 
with  intermediate  sounds""".  In  this  instrument,  then, 
the  length  of  the  pipe  would  determine  the  vowel,  and 
might  be  used  numerically  to  express  it.  Such  an  in 
strument  so  employed  would  be  a  measure  of  vowel 
quality,  and  might  be  called  a  Phthongometer. 

Our  business  at  present,  however,  is  not  with  instru 
ments  which  might  be  devised  for  measuring  sensible 
qualities,  but  with  those  which  have  been  so  used,  and 
have  thus  been  the  basis  of  the  sciences  in  which  such 
qualities  are  treated  of;  and  this  we  have  now  done  suf 
ficiently  for  our  present  purpose. 

24.  There  is  another  Idea  which,  though  hitherto 
very  vaguely  entertained,  has  had  considerable  influence 
in  the  formation,  both  of  the  sciences  spoken  of  in  the 
present  Book,  and  on  others  which  will  hereafter  come 
under  our  notice :  namely,  the  Idea  of  Polarity.  This 
Idea  will  be  the  subject  of  the  ensuing  Book.  And 
although  this  Idea  forms  a  part  of  the  basis  of  various 
other  extensive  portions  of  science,  as  Optics  and  Che 
mistry,  it  occupies  so  peculiarly  conspicuous  a  place  in 
speculations  belonging  to  what  I  have  termed  the  Mecha- 
nico-Chemical  Sciences,  (Magnetism  and  Electricity,) 
that  I  shall  designate  the  discussion  of  the  Idea  of 
Polarity  as  the  Philosophy  of  those  Sciences. 

*   Camb.  Trans.,  A"ol.  in.  p.  239. 






1.  IN  some  of  the  mechanical  sciences,  as  Magnetism 
and  Optics,  the  phenomena  are  found  to  depend  upon 
position  (the  position  of  the  magnet,  or  of  the  ray  of 
light,)  in  a  peculiar  alternate  manner.  This  dependence, 
as  it  was  first  apprehended,  was  represented  by  means 
of  certain  conceptions  of  space  and  force,  as  for  instance 
by  considering  the  two  poles  of  a  magnet.  But  in  all 
such  modes  of  representing  these  alternations  by  the 
conceptions  borrowed  from  other  ideas,  a  closer  exami 
nation  detected  something  superfluous  and  something 
defective ;  and  in  proportion  as  the  view  which  philo 
sophers  took  of  this  relation  was  gradually  purified  from 
these  incongruous  elements,  and  was  rendered  more 
general  and  abstract  by  the  discovery  of  analogous  pro 
perties  in  new  cases,  it  was  perceived  that  the  relation 
could  not  be  adequately  apprehended  without  consider 
ing  it  as  involving  a  peculiar  and  independent  Idea, 
which  we  may  designate  by  the  term  Polarity. 

We  shall  trace  some  of  the  forms  in  which  this  Idea 
has  manifested  itself  in  the  history  of  science.  In  doing 
so  we  shall  not  begin,  as  in  other  Books  of  this  work 


we  have  done,  by  speaking  of  the  notion  as  it  is  em 
ployed  in  common  use :  for  the  relation  of  polarity  is  of 
so  abstract  and  technical  a  nature,  that  it  is  not  employed, 
at  least  in  any  distinct  and  obvious  manner,  on  any 
ordinary  or  practical  occasions.  The  idea  belongs  pecu 
liarly  to  the  region  of  speculation :  in  persons  of  com 
mon  habits  of  thought  it  is  probably  almost  or  quite 
undeveloped  ;  and  even  most  of  those  whose  minds  have 
been  long  occupied  by  science,  find  a  difficulty  in  appre 
hending  it  in  its  full  generality  and  abstraction,  and 
stript  of  all  irrelevant  hypothesis. 

2.  Magnetism, — The  name  and  the  notion  of  Poles 


were  first  adopted  in  the  case  of  a  magnet.  If  we  have 
two  magnets,  their  extremities  attract  and  repel  each 
other  alternatively.  If  the  first  end  of  the  one  attract 
the  first  end  of  the  other,  it  repels  the  second  end,  and 
conversely.  In  order  to  express  this  rule  conveniently, 
the  two  ends  of  each  magnet  are  called  the  north  pole 
and  the  south  pole  respectively,  the  denominations  being 
borrowed  from  the  poles  of  the  earth  and  heavens. 
"These  poles,"  as  Gilbert  says'",  ''regulate  the  motions 
of  the  celestial  spheres  and  of  the  earth.  In  like  manner 
the  magnet  has  its  poles,  a  northern  and  a  southern  one ; 
certain  and  determined  points  constituted  by  nature  in 
the  stone,  the  primary  terms  of  its  motions  and  effects, 
the  limits  and  governors  of  many  actions  and  virtues." 

The  nature  of  the  opposition  of  properties  of  which 
we  speak  may  be  stated  thus. 

The  North  pole  of  one  magnet  attracts  the  South 
pole  of  another  magnet. 

The  North  pole  of  one  magnet  repels  the  North  pole 
of  another  magnet. 

The  South  pole  of  one  magnet  repels  the  South  pole 
of  another  magnet. 

*  DC  Matin.,  Lib.  i.  c.  iii. 

APPLICATION    OF   THE    IDEA    OF    POLARITY.      347 

The  South  pole  of  one  magnet  attracts  the  North 
pole  of  another  magnet. 

It  will  be  observed  that  the  contrariety  of  position 
which  is  indicated  by  putting  the  South  pole  for  the 
North  pole  in  either  magnet,  is  accompanied  by  the 
opposition  of  mechanical  effect  which  is  expressed  by 
changing  attraction  into  repulsion  and  repulsion  into 
attraction :  and  thus  we  have  the  general  feature  of 
polarity : — A  contrast  of  properties  corresponding  to  a 
contrast  of  positions. 

!J.  Electricity. — When  the  phenomena  of  electricity 
came  to  be  studied,  it  appeared  that  they  involved  rela 
tions  in  some  respects  analogous  to  those  of  magnetism. 

Two  kinds  of  electricity  were  distinguished,  the 
positive  and  the  negative ;  and  it  appeared  that  two 
bodies  electrized  positively  or  two  electrized  negatively, 
repelled  each  other,  like  two  north  or  two  south  magnetic 
poles ;  while  a  positively  and  a  negatively  electrized  body 
attracted  each  other,  like  the  north  and  south  poles  of 
two  magnets.  In  conductors  of  an  oblong  form,  the 
electricity  could  easily  be  made  to  distribute  itself  so 
that  one  end  should  be  positively  and  one  end  negatively 
electrized ;  and  then  such  conductors  acted  on  each  other 
exactly  as  magnets  would  do. 

But  in  conductors,  however  electrized,  there  is  no 
peculiar  point  which  can  permanently  be  considered  as 
the  pole.  The  distribution  of  electricity  in  the  conduc 
tor  depends  upon  external  circumstances :  and  thus, 
although  the  phenomena  offer  the  general  character  of 
polarity — alternative  results  corresponding  to  alternative 
positions,— they  cannot  be  referred  to  poles.  Some  other 
mode  of  representing  the  forces  must  be  adopted  than 
that  which  makes  them  emanate  from  permanent  points 
as  in  a  magnet. 

The  phenomena  of  attraction  and  repulsion  in  elec- 


trized  bodies  were  conveniently  represented  by  means  of 
the  hypothesis  of  tiro  electric  fluids,  a  positive  and  a 
negative  one,  which  were  supposed  to  be  distributed  in 
the  bodies.  Of  these  fluids,  it  was  supposed  that  each 
repelled  its  own  parts  and  attracted  those  of  the  opposite 
fluid  :  and  it  was  found  that  this  hypothesis  explained  all 
the  obvious  laws  of  electric  action.  Here  then  we  have 
the  phenomena  of  polarization  explained  by  a  new  kind 
of  machinery  : — two  opposite  fluids  distributed  in  bodies, 
and  supplying  them,  so  to  speak,  with  their  polar  forces. 
This  hypothesis  not  only  explains  electrical  attraction, 
but  also  the  electrical  spark  :  when  two  bodies,  of  which 
the  neighbouring  surfaces  are  charged  with  the  two 
opposite  fluids,  approach  near  to  each  other,  the  mutual 
attraction  of  the  fluids  becomes  more  and  more  intense, 
till  at  last  the  excess  of  fluid  on  the  one  body  breaks 
through  the  air  and  rushes  to  the  other  body,  in  a  form 
accompanied  by  light  and  noise.  When  this  transfer  has 
taken  place,  the  attraction  ceases,  the  positive  and  the 
negative  fluid  having  neutralized  each  other.  Their 
effort  was  to  unite ;  and  this  union  being  effected,  there 
is  no  longer  any  force  in  action.  Bodies  in  their  natural 
unexcited  condition  may  be  considered  as  occupied  by  a 
combination  of  the  two  fluids :  and  hence  we  see  how 
the  production  of  either  kind  of  electricity  is  necessarily 
accompanied  with  the  production  of  an  equivalent  amount 
of  the  opposite  kind. 

4.  Voltaic  Electricity. — Such  is  the  case  in  Franklinic 
electricity, — that  which  is  excited  by  the  common  elec 
trical  machine.  In  studying  Voltaic  electricity,  we  are 
led  to  the  conviction  that  the  fluid  which  is  in  a  condi 
tion  of  momentary  equilibrium  in  electrized  conductors, 
exists  in  the  state  of  current  in  the  voltaic  circuit.  And 
here  we  find  polar  relations  of  a  new  kind  existing  among 
the  forces.  Two  voltaic  currents  attract  each  other  when 


they  are  moving  in  the  same,  and  repel  each  other  when 
they  are  moving  in  opposite,  directions. 

But  we  find,  in  addition  to  these,  other  polar  rela 
tions  of  a  more  abstruse  kind,  and  which  the  supposition 
of  two  fluids  does  not  so  readily  explain.  For  instance, 
if  such  fluids  existed,  distinct  from  each  other,  it  might 
be  expected  that  it  would  be  possible  to  exhibit  one 
of  them  separate  from  the  other.  Vet  in  all  the  phe 
nomena  of  electromotive  currents,  we  attempt  in  vain 
to  obtain  one  kind  of  electricity  separately.  "I  have 
not,"  says  Mr.  Faraday'",  "been  able  to  find  a  single 
fact  which  could  be  adduced  to  prove  the  theory  of 
two  electricities  rather  than  one,  in  electric  currents; 
or,  admitting  the  hypothesis  of  two  electricities,  have 
I  been  able  to  perceive  the  slightest  grounds  that  one 
electricity  can  be  more  powerful  than  the  other, — or 
that  it  can  be  present  without  the  other, — or  that  it 
can  be  varied  or  in  the  slightest  degree  affected  without 
a  corresponding  variation  in  the  other."  "  Thus,"  he 
adds,  "  the  polar  character  of  the  powers  is  rigorous  and 
complete."  Thus,  we  too  may  remark,  all  the  super 
fluous  and  precarious  parts  gradually  drop  off  from  the 
hypothesis  which  we  devise  in  order  to  represent  polar 
phenomena;  and  the  abstract  notion  of  polarity — of  equal 
and  opposite  powers  called  into  existence  by  a  com 
mon  condition — remains  unincumbered  with  extraneous 

5.  Light. — Another  very  important  example  of  the 
application  of  the  idea  of  polarity  is  that  supplied  by  the 
discovery  of  the  polarization  of  light.  A  ray  of  light 
may,  by  various  processes,  be  modified,  so  that  it  has  dif 
ferent  properties  according  to  its  different  sides,  although 
this  difference  is  not  perceptible  by  any  common  effects. 
If,  for  instance,  a  ray  thus  modified,  pass  perpendicularly 

*   Rexearches,  ;">](>. 


through  a  circular  glass,  and  fall  upon  the  eye,  we  may 
turn  the  glass  round  and  round  its  frame,  and  we  shall 
made  no  difference  in  the  brightness  of  the  spot  which 
we  see.  But  if,  instead  of  a  glass,  we  look  through  a 
longitudinal  slice  of  tourmaline,  the  spot  is  alternately 
dark  and  bright  as  we  turn  the  crystal  through  successive 
quadrants.  Here  we  have  a  contrast  of  properties  (dark 
and  bright)  corresponding  to  a  contrast  of  positions,  (the 
position  of  a  line  east  and  west  being  contrasted  with 
the  position  north  and  south,)  which,  as  we  have  said,  is 
the  general  character  of  polarity.  It  was  with  a  view  of 
expressing  this  character  that  the  term  polarization  was 
originally  introduced.  Mains  was  forced  by  his  disco 
veries  into  the  use  of  this  expression.  "  We  find,"  he 
says,  in  1811,  "that  light  acquires  properties  which  are 
relative  only  to  the  sides  of  the  ray, — which  are  the  same 
for  the  north  and  south  sides  of  the  ray,  (using  the 
points  of  the  compass  for  description's  sake  only,)  and 
which  are  different  when  we  go  from  the  north  and  south 
to  the  east  or  to  the  west  sides  of  the  ray.  I  shall  give 
the  name  of  poles  to  these  sides  of  the  ray,  and  shall  call 
polarization  the  modification  which  gives  to  light  these 
properties  relative  to  these  poles.  I  have  put  off  hitherto 
the  admission  of  this  term  into  the  description  of  the 
physical  phenomena  with  which  we  have  to  do :  I  did 
not  dare  to  introduce  it  into  the  Memoirs  in  which  I 
published  my  last  observations :  but  the  variety  of  forms 
in  which  this  new  phenomenon  appears,  and  the  difficulty 
of  describing  them,  compel  me  to  admit  this  new  expres 
sion  ;  which  signifies  simply  the  modification  which  light 
has  undergone  in  acquiring  new  properties  which  are  not 
relative  to  the  direction  of  the  ray,  but  only  to  its  sides 
considered  at  right  angles  to  each  other,  and  in  a  plane 
perpendicular  to  its  direction." 

The  theory  which  represents  light  as  an  emission  of 


particles  was  in  vogue  at  the  time  when  Mains  published 
his  discoveries;  and  some  of  his  followers  in  optical 
research  conceived  that  the  phenomena  which  he  thus 
described  rendered  it  necessary  to  ascribe  poles  and  an 
axis  to  each  particle  of  light.  On  this  hypothesis,  light 
would  be  polarized  when  the  axes  of  all  the  particles 
were  in  the  same  direction :  and,  making  such  a  suppo 
sition,  it  may  easily  be  conceived  capable  of  transmission 
through  a  crystal  whose  axis  is  parallel  to  that  of  the 
luminous  particles,  and  intransmissible  when  the  axis  of 
the  crystal  is  in  a  position  transverse  to  that  of  the  par 

The  hypothesis  of  particles  possessing  poles  is  a  rude 
and  arbitrary  assumption,  in  this  as  in  other  cases ;  but 
it  serves  to  convey  the  general  notion  of  polarity,  which 
is  the  essential  feature  of  the  phenomena.  The  term 
"polarization  of  light"  has  sometimes  been  complained 
of  in  modern  times  as  hypothetical  and  obscure.  But  the 
real  cause  of  obscurity  was,  that  the  Idea  of  Polarity  was, 
till  lately,  very  imperfectly  developed  in  men's  minds. 
As  we  have  seen,  the  general  notion  of  polarity, — oppo 
site  properties  in  opposite  directions, — exactly  describes 
the  character  of  the  optical  phenomena  to  which  the 
term  is  applied. 

It  is  to  be  recollected  that  in  optics  we  never  speak 
of  the  poles,  but  of  the  plane  of  polarization  of  a  ray.  The 
word  sides,  which  Newton  and  Mains  have  used,  neither 
of  them  appears  to  have  been  satisfied  with ;  Newton,  in 
employing  it,  had  recourse  to  the  strange  Gallicism  of 
speaking  of  the  coast  of  usual  and  of  unusual  refraction 
of  a  crystal. 

The  modern  theory  of  optics  represents  the  plane  of 
polarization  of  light  as  depending,  not  on  the  position  in 
which  the  axes  of  the  luminifcrous  particles  lie,  but  on 
the  direction  of  those  transverse  vibrations  in  which  light 


consists.  This  theory  is,  as  we  have  stated  in  the  His 
tory,  recommended  by  an  extraordinary  series  of  suc 
cesses  in  accounting  for  the  phenomena.  And  this 
hypothesis  of  transverse  vibrations  shows  us  another 
mechanical  mode,  (besides  the  hypothesis  of  particles 
with  axes,)  by  which  we  may  represent  the  polarity  of  a 
ray.  But  we  may  remark  that  the  general  notion  of 
polarity,  as  applied  to  light  in  such  cases,  would  subsist, 
even  if  the  undulatory  theory  were  rejected.  The  idea 
is,  as  we  have  before  said,  independent  of  all  hypothetical 

I  need  not  here  refer  to  the  various  ways  in  which 
light  may  be  polarized,  as,  for  instance,  by  being  reflected 
from  the  surface  of  water  or  of  glass  at  certain  angles,  by 
being  transmitted  through  crystals,  and  in  other  ways. 
In  all  cases  the  modification  produced,  the  polarization, 
is  identically  the  same  property.  Nor  need  I  mention 
the  various  kinds  of  phenomena  which  appear  as  contrasts 
in  the  result ;  for  these  are  not  merely  light  and  dark,  or 
white  and  black,  but  red  and  green,  and  generally,  a 
colour  and  its  complementary  colour,  exhibited  in  many 
complex  and  varied  configurations.  These  multiplied 
modes  in  which  polarized  light  presents  itself  add  nothing 
to  the  original  conception  of  polarization :  and  I  shall 
therefore  pass  on  to  another  subject. 

G.  Crystallization. — Bodies  which  are  perfectly  crys 
tallized  exhibit  the  most  complete  regularity  and  sym 
metry  of  form ;  and  this  regularity  not  only  appears  in 
their  outward  shape,  but  pervades  their  whole  texture, 
and  manifests  itself  in  their  cleavage,  their  transparency, 
and  in  the  uniform  and  determinate  optical  properties 
which  exist  in  every  part,  even  the  smallest  fragment  of 
the  mass.  If  we  conceive  crystals  as  composed  of  par 
ticles,  we  must  suppose  these  particles  to  be  arranged  in 
the  most  regular  manner ;  for  example,  if  we  suppose 


each  particle  to  have  an  axis,  we  must  suppose  all  these 
axes  to  be  parallel ;  for  the  direction  of  the  axis  of  the 
particles  is  indicated  by  the  physical  and  optical  pro 
perties  of  the  crystal,  and  therefore  this  direction  must 
be  the  same  for  every  portion  of  the  crystal.  This 
parallelism  of  the  axes  of  the  particles  may  be  con 
ceived  to  result  from  the  circumstance  of  each  particle 
having  poles,  the  opposite  poles  attracting  each  other. 
In  virtue  of  forces  acting  as  this  hypothesis  assumes,  a 
collection  of  small  magnetic  particles  would  arrange 
themselves  in  parallel  positions ;  and  such  a  collection  of 
magnetic  particles  offers  a  sort  of  image  of  a  crystal. 
Thus  we  are  led  to  conceive  the  particles  of  crystals  as 
polarized,  and  as  determined  in  their  crystalline  positions 
by  polar  forces.  This  mode  of  apprehending  the  consti 
tution  of  crystals  has  been  adopted  by  some  of  our  most 
eminent  philosophers.  Thus  Berzelius  says""',  "It  is  de 
monstrated,  that  the  regular  forms  of  bodies  presuppose 
an  effort  of  their  atoms  to  touch  each  other  by  preference 
in  certain  points ;  that  is,  they  are  founded  upon  a  Pola 
rity  ;" — he  adds,  "  a  polarity  which  can  be  no  other  than 
an  electric  or  magnetic  polarity."  In  this  latter  clause 
we  have  the  identity  of  different  kinds  of  polarity 
asserted ;  a  principle  which  we  shall  speak  of  in  the 
next  chapter.  But  we  may  remark,  that  even  without 
dwelling  upon  this  connexion,  any  notion  which  we  can 
form  of  the  structure  of  crystals  necessarily  involves  the 
idea  of  polarity.  Whether  this  polarity  necessarily  re 
quires  us  to  believe  crystals  to  be  composed  of  atoms 
which  exert  an  effort  to  touch  each  other  in  certain  points 
by  preference,  is  another  question.  And,  in  agreement 
with  what  has  been  said  respecting  other  kinds  of  polarity, 
we  shall  probably  find,  on  a  more  profound  examination 
of  the  subject,  that  while  the  idea  of  polarity  is  essential, 

::"   Essay  on  the  T/ieori/  of  Chemical  Properties,  15520,  j>.  \\\\. 
VOL.    I.       W.  P.  A  A 


the  machinery  by  which  it  is  thus  expressed  is  precarious 
and  superfluous. 

7.  Chemical  Affinity. — We  shall  have,  in  the  next 
Book,  to  speak  of  Chemical  Affinity  at  some  length ;  but 
since  the  ultimate  views  to  which  philosophers  have  been 
led,  induce  them  to  consider  the  forces  of  affinity  as 
polar  forces,  we  must  enumerate  these  among  the  exam 
ples  of  polarity.  In  chemical  processes,  opposites  tend 
to  unite,  and  to  neutralize  each  other  by  their  union. 
Thus  an  acid  or  an  alkali  combine  with  vehemence,  and 
form  a  compound,  a  neutral  salt,  which  is  neither  acid 
nor  alkaline. 

This  conception  of  contrariety  and  mutual  neutraliza 
tion,  involves  the  idea  of  polarity.    In  the  conception,  as 
entertained  by  the  earlier  chemists,  the  idea  enters  very 
obscurely :  but  in  the  attempts  which  have  more  recently 
been  made  to  connect  this  relation  (of  acid  and  base,)  with 
other  relations,  the  chemical  elements  have  been  conceived 
as  composed  of  particles  which  possess  poles ;  like  poles 
repelling,  and  unlike  attracting  each  other,  as  they  do  in 
magnetic  and  electric  phenomena.     This  is,  however,  a 
rude  and  arbitrary  way  of  expressing  polarity,  and,  as  may 
be  easily  shown,  involves  many  difficulties  which  do  not 
belong  to  the  idea  itself.     Mr.  Faraday,  who  has  been 
led  by  his  researches  to  a  conviction  of  the  polar  nature 
of  the  forces  of  chemical  affinity,   has  expressed    their 
character  in  a  more  general  manner,  and  without  any  of 
the  machinery  of  particles  indued  with  poles.     Accord 
ing  to  his  view,  chemical  synthesis  and   analysis  must 
always  be  conceived  as  taking  place  in  virtue  of  equal 
and  opposite  forces,  by  which  the  particles  are  united  or 
separated.     These  forces,   by  the   very  circumstance  of 
their  being  polar,  may  be  transferred  from  point  to  point. 
For  if  we  conceive  a  string  of  particles,  and  if  the  positive 
force  of  the  first  particle  be  liberated  and  brought  into 


action,  its  negative  force  also  must  be  set  free :  this 
negative  force  neutralizes  the  positive  force  of  the  next 
particle,  and  therefore  the  negative  force  of  this  particle 
(before  employed  in  neutralizing  its  positive  force,)  is  set 
free :  this  is  in  the  same  way  transferred  to  the  next 
particle,  and  so  on.  And  thus  we  have  a  positive  force 
active  at  one  extremity  of  a  line  of  particles,  correspond 
ing  to  a  negative  force  at  the  other  extremity,  all  the 
intermediate  particles  reciprocally  neutralizing  each 
other's  action.  This  conception  of  the  transfer  of  chemi 
cal  action  was  indeed  at  an  earlier  period  introduced  by 
Grotthus"-",  and  confirmed  by  Davy.  But  in  Mr.  Fara 
day's  hands  we  see  it  divested  of  all  that  is  superfluous, 
and  spoken  of,  not  as  a  line  of  particles,  but  as  *'  an  axis 
of  power,  having  [at  every  point,]  contrary  forces,  ex 
actly  equal,  in  opposite  directions." 

8.  General  Remarks. — Thus,  as  we  see,  the  notion 
of  polarity  is  applicable  to  many  large  classes  of  phe 
nomena.  Yet  the  idea  in  a  distinct  and  general  form  is 
only  of  late  growth  among  philosophers.  It  has  gra 
dually  been  abstracted  and  refined  from  many  extraneous 
hypotheses  which  were  at  first  supposed  to  be  essential 
to  it.  We  have  noticed  some  of  these  hypotheses ; — as 
the  poles  of  a  body;  the  poles  of  the  particles  of  a  fluid  ; 
two  opposite  fluids;  a  single  fluid  in  excess  and  defect; 
transverse  vibrations.  To  these  others  might  be  added. 
Thus  Dr.  Proutf  assumes  that  the  polarity  of  molecules 
results  from  their  rotation  on  their  axes,  the  opposite 
motions  of  contiguous  molecules  being  the  cause  of 
opposite  (positive  and  negative)  polarities. 

But  none  of  these  hypotheses  can  be  proved  by  the 
fact  of  polarity  alone ;  and  they  have  been  in  succession 
rejected  when  they  had  been  assumed  on  that  ground. 

*  DUMAS,  Lcqons  sur  la  Philosophy  Chimiqite,  p.  401. 
t   Bridgcwater  Treatixe,  p.  .r>.~>i). 

A  A  "2 


Thus  Davy,  in  18*20,  speaking  of  chemical  forces  says*, 
•'  In  assuming  the  idea  of  two  ethereal,  subtile,  elastic 
fluids,  attractive  of  the  particles  of  each  other,  and 
repulsive  as  to  their  own  particles,  capable  of  combining 
in  different  proportions  with  bodies,  and  according  to 
their  proportions  giving  them  their  specific  qualities  and 
rendering  them  equivalent  masses,  it  would  be  natural 
to  refer  the  action  of  the  poles  to  the  repulsions  of  the 
substances  combined  with  the  excess  of  one  fluid,  and 
the  attractions  of  those  united  to  the  excess  of  the  other 
fluid;  and  a  history  of  the  phenomena,  not  unsatisfactory 
to  the  reason,  might  in  this  way  be  made  out.  But  as 
it  is  possible  likewise  to  take  an  entirely  different  view 
of  the  subject,  on  the  idea  of  the  dependence  of  the 
results  upon  the  primary  attractive  powers  of  the  parts 
of  the  combination  on  a  single  subtile  fluid,  I  shall  not 
enter  into  any  discussion  on  this  obscure  part  of  the 
theory."  Which  of  these  theories  will  best  represent  the 
case,  will  depend  upon  the  consideration  of  other  facts, 
in  combination  with  the  polar  phenomena,  as  we  see  in 
the  history  of  optical  theory.  In  like  manner  Mr. 
Faraday  proved  by  experiment  f  the  errour  of  all  theories 
which  ascribe  electro-chemical  decomposition  to  the 
attraction  of  the  poles  of  the  voltaic  battery. 

In  order  that  they  may  distinctly  image  to  them 
selves  the  idea  of  polarity,  men  clothe  it  in  some  of 
the  forms  of  machinery  above  spoken  of;  yet  every  new 
attempt  shows  them  the  unnecessary  difficulties  in  which 
they  thus  involve  themselves.  But  on  the  other  hand 
it  is  difficult  to  apprehend  this  idea  divested  of  all 
machinery ;  and  to  entertain  it  in  such  a  form  that  it 
shall  apply  at  the  same  time  to  magnetism  and  elec 
tricity,  galvanism  and  chemistry,  crystalline  structure 
and  light.  The  Idea  of  Polarity  becomes  most  pure  and 

-   Phil.  7V.,  182(5,  p.  41.1.  I    Itwarc/ies,  p.  495,  &c. 


genuine,  when  we  entirely  reject  the  conception  of  Poles, 
as  Faraday  has  taught  us  to  do  in  considering  electro 
chemical  decomposition ;  but  it  is  only  by  degrees  and 
by  effort  that  we  can  reach  this  point  of  abstraction  and 

9.  There  is  one  other  remark  which  we  may  here 
make.  It  was  a  maxim  commonly  received  in  the  ancient 
schools  of  philosophy,  that  "  like  attracts  like :"  but  as 
we  have  seen,  the  universal  maxim  of  polar  phenomena 
is,  that  like  repels  like,  and  attracts  unlike.  The  north 
pole  attracts  the  south  pole,  the  positive  fluid  attracts 
the  negative  fluid ;  opposite  elements  rush  together ; 
opposite  motions  reduce  each  other  to  rest.  The  per 
manent  and  stable  course  of  things  is  that  which  results 
from  the  balance  and  neutralization  of  contrary  ten 
dencies.  Nature  is  constantly  labouring  after  repose  by 
the  effect  of  such  tendencies ;  and  so  far  as  polar  forces 
enter  into  her  economy,  she  seeks  harmony  by  means  of 
discord,  and  unity  by  opposition. 

Although  the  Idea  of  Polarity  is  as  yet  somewhat 
vague  and  obscure,  even  in  the  minds  of  the  cultivators 
of  physical  science,  it  has  nevertheless  given  birth  to 
some  general  principles  which  have  been  accepted  as 
evident,  and  have  had  great  influence  on  the  progress 
of  science.  These  we  shall  now  consider. 


1.  IT  has  appeared  in  the  preceding  chapter  that  in 
cases  in  which  the  phenomena  suggest  to  us  the  idea  of 
polarity,  we  are  also  led  to  assume  some  material  ma 
chinery  as  the  mode  in  which  the  polar  forces  are  exerted. 


We  assume,  for  instance,  globular  particles  which  possess 
poles,  or  the  vibrations  of  a  fluid,  or  two  fluids  attract 
ing  each  other ;  in  every  case,  in  short,  some  hypothesis 
by  which  the  existence  and  operation  of  the  polarity  is 
embodied  in  geometrical  and  mechanical  properties  of  a 
medium ;  nor  is  it  possible  for  us  to  avoid  proceeding 
upon  the  conviction  that  some  such  hypothesis  must  be 
true ;  although  the  nature  of  the  connexion  between 
the  mechanism  and  the  phenomena  must  still  be  inde 
finite  and  arbitrary. 

But  since  each  class  of  polar  phenomena  is  thus 
referred  to  an  ulterior  cause,  of  which  we  know  no  more 
than  that  it  has  a  polar  character,  it  follows  that  different 
polarities  may  result  from  the  same  cause  manifesting 
its  polar  character  under  different  aspects.  Taking,  for 
example,  the  hypothesis  of  globular  particles,  if  elec 
tricity  result  from  an  action  dependent  upon  the  poles 
of  each  globule,  magnetism  may  depend  upon  an  action 
in  the  equator  of  each  globule;  or  taking  the  supposition 
of  transverse  vibrations,  if  polarized  light  result  directly 
from  such  vibrations,  crystallization  may  have  reference 
to  the  axes  of  the  elasticity  of  the  medium  by  which  the 
vibrations  are  rendered  transverse, — so  far  as  the  polar 
character  only  of  the  phenomena  is  to  be  accounted  for. 
I  say  this  may  be  so,  in  so  far  only  as  the  polar  cha 
racter  of  the  phenomena  is  concerned ;  for  whether  the 
relation  of  electricity  to  magnetism,  or  of  crystalline 
forces  to  light,  can  really  be  explained  by  such  hypo 
theses,  remains  to  be  determined  by  the  facts  themselves. 
But  since  the  first  necessary  feature  of  the  hypothesis 
is,  that  it  shall  give  polarity,  and  since  an  hypothesis 
which  does  this,  may,  by  its  mathematical  relations,  give 
polarities  of  different  kinds  and  in  different  directions, 
any  two  co-existent  kinds  of  polarity  may  result  from 
the  same  cause,  manifesting  itself  in  various  manners. 


The  conclusion  to  which  we  are  led  by  these  general 
considerations  is,  that  two  co-existing  classes  of  polar 
phenomena  may  be  effects  of  the  same  cause.  But  those 
who  have  studied  such  phenomena  more  deeply  and 
attentively  have,  in  most  or  in  all  cases,  arrived  at  the 
conviction  that  the  various  kinds  of  polarity  in  such 
cases  must  be  connected  and  fundamentally  identical. 
As  this  conviction  has  exercised  a  great  influence,  both 
upon  the  discoveries  of  new  facts  and  upon  the  theore 
tical  speculations  of  modern  philosophers,  and  has  been 
put  forward  by  some  writers  as  a  universal  principle  of 
science,  I  will  consider  some  of  the  cases  in  which  it  has 
been  thus  applied. 

'2.  Connexion  of  Magnetic  and  Electric  Polarity. — 
The  polar  phenomena  of  electricity  and  magnetism  are 
clearly  analogous  in  their  laws:  and  obvious  facts  showed 
at  an  early  period  that  there  was  some  connexion  be 
tween  the  two  agencies.  Attempts  were  made  to  esta 
blish  an  evident  and  definite  relation  between  the  two 
kinds  of  force,  which  attempts  proceeded  upon  the  prin 
ciple  now  under  consideration ; — namely,  that  in  such 
cases,  the  two  kinds  of  polarity  must  be  connected.  Pro 
fessor  (Ersted,  of  Copenhagen,  was  one  of  those  who 
made  many  trials  founded  upon  this  conviction :  yet  all 
these  were  long  unsuccessful.  At  length,  in  1820,  he 
discovered  that  a  galvanic  current,  passing  at  right  angles 
near  to  a  magnetic  needle,  exercises  upon  it  a  powerful 
deflecting  force.  The  connexion  once  detected  between 
magnetism  and  galvanism  was  soon  recognized  as  con 
stant  and  universal.  It  was  represented  in  different 
hypothetical  modes  by  different  persons ;  some  consider 
ing  the  galvanic  current  as  the  primitive  axis,  and  the 
magnet  as  constituted  of  galvanic  currents  passing  round 
it  at  right  angles  to  the  magnetic  axis;  while  others 
conceived  the  magnetic  axis  as  the  primitive  one,  and 


the  electric  current  as  implying  a  magnetic  current 
round  the  wire.  So  far  as  many  of  the  general  relations 
of  these  two  kinds  of  force  were  concerned,  either  mode 
of  representation  served  to  express  them ;  and  thus  the 
assumption  that  the  two  polarities,  the  magnetic  and 
the  electric,  were  fundamentally  identical,  was  verified, 
so  far  as  the  phenomena  of  magnetic  attraction,  and  the 
like,  were  concerned. 

I  need  not  here  mention  how  this  was  further  con 
firmed  by  the  experiments  in  which,  by  means  of  the 
forces  thus  brought  into  view,  a  galvanic  wire  was  made 
to  revolve  round  a  magnet,  and  a  magnet  round  a  gal 
vanic  wire  ; — in  which  artificial  magnets  were  constructed 
of  coils  of  galvanic  wire  ; — and  finally,  in  which  the  gal 
vanic  spark  was  obtained  from  the  magnet.  The  identity 
which  sagacious  speculators  had  divined  even  before  it 
was  discovered,  and  which  they  had  seen  to  be  universal 
as  soon  as  it  was  brought  to  light,  was  completely  mani 
fested  in  every  imaginable  form. 

The  relation  of  the  electric  and  magnetic  polarities 
was  found  to  be,  that  they  were  transverse  to  each 
other,  and  this  relation  exhibited  under  various  condi 
tions  of  form  and  position  of  the  apparatus,  gave  rise  to 
very  curious  and  unexpected  perplexities.  The  degree 
of  complication  which  this  relation  may  occasion,  may  be 
judged  of  from  the  number  of  constructions  and  modes 
of  conception  offered  by  CErsted,  Wollaston,  Faraday, 
and  others,  for  the  purpose  of  framing  a  technical  memory 
of  the  results.  The  magnetic  polarity  gives  us  the  north 
and  south  poles  of  the  needle ;  the  electric  polarity 
makes  the  current  positive  and  negative;  and  these  pairs 
of  opposites  are  connected  by  relations  of  situation,  as 
above  and  below,  right  and  left ;  and  give  rise  to  the 
resulting  motion  of  the  needle  one  way  or  the  other. 

3.    Ampere,  by  framing  his  hypotheses  of  the  action 


of  voltaic  currents  and  the  constitution  of  magnets, 
reduced  all  these  technical  rules  to  rigorous  deductions 
from  one  general  principle.  And  thus  the  vague  and 
obscure  persuasion  that  there  must  be  some  connexion 
between  electricity  and  magnetism,  so  long  an  idle  and 
barren  conjecture,  was  unfolded  into  a  complete  theory, 
according  to  which  magnetic  and  electromotive  actions 
are  only  two  different  manifestations  of  the  same  forces; 
and  all  the  above-mentioned  complex  relations  of  pola 
rities  are  reduced  to  one  single  polarity,  that  of  the 
electro-dynamic  current. 

4.  As  the  idea  of  polarity  was  thus  firmly  established 
and  clearly  developed,  it  became  an  instrument  of  rea 
soning.  Thus  it  led  Ampere  to  maintain  that  the  original 
or  elementary  forces  in  electro-dynamic  action  could  not 
be  as  M.  Biot  thought  they  were,  a  statical  couple,  but 
must  be  directly  opposite  to  each  other.  The  same  idea 
enabled  Mr.  Faraday  to  carry  on  with  confidence  such 
reasonings  as  the  following'"" :  "  No  other  known  power 
has  like  direction  with  that  exerted  between  an  electric 
current  and  a  magnetic  pole ;  it  is  tangential,  while  all 
other  forces  acting  at  a  distance  are  direct.  Hence  if  a 
magnetic  pole  on  one  side  of  a  revolving  plate  follow 
its  course  by  reason  of  its  obedience  to  the  tangential 
force  exerted  upon  it  by  the  very  current  of  electricity 
which  it  has  itself  caused ;  a  similar  pole  on  the  other 
side  of  the  plate  should  immediately  set  it  free  from  this 
force ;  for  the  currents  which  have  to  be  formed  by  the 
two  poles  are  in  contrary  directions."  And  in  Article 
1114  of  his  Researches,  the  same  eminent  philosopher 
infers  that  if  electricity  and  magnetism  are  considered 
as  the  results  of  a  peculiar  agent  or  condition,  exerted 
in  determinate  directions  perpendicular  to  each  other, 
one  must  be  by  some  means  convertible  into  the  other; 
"  Researches.  244. 


and  this  he  was  afterwards  able  to  prove  to  be  the  case 
in  fact. 

Thus  the  principle  that  the  co-existent  polarities  of 
magnetism  and  electricity  are  connected  and  fundamen 
tally  identical,  is  not  only  true,  but  is  far  from  being- 
cither  vague  or  barren.  It  has  been  a  fertile  source 
both  of  theories  which  have,  at  present,  a  very  great  pro 
bability,  and  of  the  discovery  of  new  and  striking  facts. 
We  proceed  to  consider  other  similar  cases. 

5.  Connexion  of  Electrical  and  Chemical  Polari 
ties. — The  doctrine  that  the  chemical  forces  by  which 
the  elements  of  bodies  are  held  together  or  separated, 
are  identical  with  the  polar  forces  of  electricity,  is  a 
great  discovery  of  modern  times ;  so  great  and  so  recent, 
indeed,  that  probably  men  of  science  in  general  have 
hardly  yet  obtained  a  clear  view  and  firm  hold  of  this 
truth.  This  doctrine  is  now,  however,  entirely  esta 
blished  in  the  minds  of  the  most  profound  and  philoso 
phical  chemists  of  our  time.  The  complete  developemerit 
and  confirmation  of  this  as  of  other  great  truths,  was 
preceded  by  more  vague  and  confused  opinions  gradu 
ally  tending  to  this  point;  and  the  progress  of  thought 
and  of  research  was  impelled  and  guided,  in  this  as  in 
similar  cases,  by  the  persuasion  that  these  co-existent 
polarities  could  not  fail  to  be  closely  connected  with 
each  other.  While  the  ultimate  and  exact  theory  to 
which  previous  incomplete  and  transitory  theories  tended 
is  still  so  new  and  so  unfamiliar,  it  must  needs  be  a 
matter  of  difficulty  and  responsibility  for  a  common 
reader  to  describe  the  steps  by  which  truth  has  advanced 
from  point  to  point.  I  shall,  therefore,  in  doing  this, 
guide  myself  mainly  by  the  historical  sketches  of  the 
progress  of  this  great  theory,  which,  fortunately  for  us, 
have  been  given  us  by  the  two  philosophers  who  have 


played  by  far  the  most  important  parts  in  the  discovery, 
Davy  and  Faraday. 

It  will  be  observed  that  we  are  concerned  here  with 
the  progress  of  theory,  and  not  of  experiment,  except  so 
far  as  it  is  confirmatory  of  theory.  In  Davy's  Memoir* 
of  1826,  on  the  Relations  of  Electrical  and  Chemical 
Changes,  he  gives  the  historical  details  to  which  I  have 
alluded.  Already  in  1802  he  had  conjectured  that  all 
chemical  decompositions  might  be  polar.  In  1806  he 
attempted  to  confirm  this  conjecture,  and  succeeded,  to 
his  own  satisfaction,  in  establishing  +  that  the  combina 
tions  and  decompositions  by  electricity  were  referable 
to  the  law  of  electrical  attractions  and  repulsions ;  and 
advanced  the  hypothesis  (as  he  calls  it,)  that  chemical 
and  electrical  attractions  were  produced  by  the  same 
cause,  acting  in  one  case  on  particles,  in  the  other  on 
masses.  This  hypothesis  was  most  strikingly  confirmed 
by  the  author's  being  able  to  use  electrical  agency  as  a 
more  powerful  means  of  chemical  decomposition  than 
any  which  had  yet  been  applied.  "  Believing,"  he  adds, 
"that  our  philosophical  systems  are  exceedingly  im 
perfect,  I  never  attached  much  importance  to  this  hypo 
thesis;  but  having  formed  it  after  a  copious  induction 
of  facts,  and  having  gained  by  the  application  of  it  a 
number  of  practical  results,  and  considering  myself  as 
much  the  author  of  it  as  I  was  of  the  decomposition  of 
the  alkalies,  and  having  developed  it  in  an  elementary 
work  as  far  as  the  present  state  of  chemistry  seemed  to 
allow,  I  have  never,"  he  says  "criticized  or  examined 
the  manner  in  which  different  authors  have  adopted  or 
explained  it,  contented,  if  in  the  hands  of  others,  it 
assisted  the  arrangements  of  chemistry  or  mineralogy, 
or  became  an  instrument  of  discovery."  When  the  doc 
trine  had  found  an  extensive  acceptance  among  chemists, 
*  Phil.  Trans.,  1826,  p.  383.  *  Ibid.,  p.  380. 


attempts  were  made  to  show  that  it  had  been  asserted 
by  earlier  writers :  and  though  Davy  justly  denies  all 
value  to  these  pretended  anticipations,  they  serve  to 
show,  however  dimly,  the  working  of  that  conviction  of 
the  connexion  of  co-existent  properties  which  all  along 
presided  in  men's  minds  during  this  course  of  investi 
gation.  "  Hitter  and  Winterl  have  been  quoted,"  Davy 
says*,  "•among  other  persons,  as  having  imagined  or 
anticipated  the  relation  between  electrical  powers  and 
chemical  affinities  before  the  discovery  of  the  pile  of 
Volta.  But  whoever  will  read  with  attention  Ritter's 
'  Evidence  that  Galvanic  action  exists  in  organized 
nature,'  and  Winter's  Prolusiones  ad  Chemiam  sceculi 
decimi  noni,  will  find  nothing  to  justify  this  opinion." 
He  then  refers  to  the  Queries  of  Xewton  at  the  end  of 
his  Optics.  "  These,"  he  says,  "  contain  more  grand  and 
speculative  views  that  might  be  brought  to  bear  upon 
this  question  than  any  found  in  the  works  of  modern 
electricians ;  but  it  is  very  unjust  to  the  experimentalists 
who  by  the  laborious  application  of  new  instruments, 
have  discovered  novel  facts  and  analogies,  to  refer  them 
to  any  such  suppositions  as  that  all  attractions,  chemical, 
electrical,  magnetical,  and  gravitative,  may  depend  upon 
the  same  cause."  It  is  perfectly  true,  that  such  vague 
opinions,  though  arising  from  that  tendency  to  generalize 
which  is  the  essence  of  science,  are  of  no  value  except 
so  far  as  they  are  both  rendered  intelligible,  and  con 
firmed  by  experimental  research. 

The  phenomena  of  chemical  decomposition  by  means 
of  the  voltaic  pile,  however,  led  other  persons  to  views 
very  similar  to  those  of  Davy.  Thus  Grotthus  in  1805  f 
published  an  hypothesis  of  the  same  kind.  "  The  pile  of 
Volta,"  he  says,  "  is  an  electrical  magnet,  of  which  each 
element,  that  is,  each  pair  of  plates,  has  a  positive  and  a 

*  Phil.  Trans.,  1826.  p.  384.  t  Ann.  C/iim..  Lxviii.  54. 


negative  pole.  The  consideration  of  this  polarity  sug 
gested  to  me  the  idea  that  a  similar  polarity  may  come 
into  play  between  the  elementary  particles  of  water 
when  acted  upon  by  the  same  electrical  agent ;  and  I 
avow  that  this  thought  was  for  me  a  flash  of  light." 

0.  The  thought,  however,  though  thus  brought  into 
being,  was  very  far  from  being  as  yet  freed  from  vague 
ness,  superfluities,  and  errours.  I  have  elsewhere  noticed* 
Faraday's  remark  on  Davy's  celebrated  Memoir  of  1806; 
that  "  the  mode  of  action  by  which  the  effects  take  place 
is  stated  very  generally,  so  generally,  indeed,  that  pro 
bably  a  dozen  precise  schemes  of  electro-chemical  action 
might  be  drawn  up,  differing  essentially  from  each  other, 
yet  all  agreeing  with  the  statement  there  given."  When 
Davy  and  others  proceeded  to  give  a  little  more  defi- 
niteness  and  precision  to  the  statement  of  their  views, 
they  soon  introduced  into  the  theory  features  which  it 
was  -  afterwards  found  necessary  to  abandon.  Thusf 
both  Davy,  Grotthus,  Riffault,  and  Chompre,  ascribed 
electrical  decomposition  to  the  action  of  the  poles,  and 
some  of  them  even  pretended  to  assign  the  proportion 
in  which  the  force  of  the  pole  diminishes  as  the  distance 
from  it  increases.  Faraday,  as  I  have  already  stated, 
showed  that  the  polarity  must  be  considered  as  residing 
not  only  in  what  had  till  then  been  called  the  poles, 
but  at  every  point  of  the  circuit.  He  ascribed  j  electro 
chemical  decomposition  to  internal  forces,  residing  in 
the  particles  of  the  matter  under  decomposition,  not  to 
external  forces,  exerted  by  the  poles.  Hence  he  shortly 
afterwards  §  proposed  to  reject  the  word  poles  altogether, 
and  to  employ  instead,  the  term  electrode,  meaning  the 

*   Hist.  Ind.  Sci.,  B.  xiv.  c.  ix.  sect.  1. 

t  See  Faraday's  Historical  Sketch,  Researches,  4H1 — 402. 

+  Art  524. 

§   In  1834.      l.lcventh  Series  of  Researches.      Art.  (502. 


doors  or  passages  (of  whatever  surface  formed,)  by  which 
the  decomposed  elements  pass  out.  What  have  been 
called  the  positive  and  negative  poles  he  further  termed 
the  anode  and  cathode  \  and  he  introduced  some  other 
changes  in  nomenclature  connected  with  these.  He 
then,  as  I  have  related  in  the  History"",  invented  the 
Volta-electrometer,  which  enabled  him  to  measure  the 
quantity  of  voltaic  action,  and  this  he  found  to  be  iden 
tical  with  the  quantity  of  chemical  affinity;  and  he  was 
thus  led  to  the  clearest  view  of  the  truth  towards  which 
he  and  his  predecessors  had  so  long  been  travelling, 
that  electrical  and  chemical  forces  are  identical!. 

7.  It  will,  perhaps,  be  said  that  this  beautiful  train 
of  discovery  was  entirely  due  to  experiment,  and  not  to 
any  a  priori  conviction  that  co-existent  polarities  must 
be  connected.     I  trust  I  have    sufficiently  stated   that 
such  an  a  priori  principle  could  not  be  proved,  nor  even 
understood,  without  a  most   laborious  and  enlightened 
use  of  experiment ;    but  yet  I  think  that  the  doctrine 
when  once  fully  unfolded,  exhibited  clearly,  and  estab 
lished  as  true,  takes  possession  of  the  mind  with  a  more 
entire    conviction  of  its   certainty  and  universality,  in 

t/  V   * 

virtue  of  the  principle  we  are  now  considering.  When 
the  theory  has  assumed  so  simple  a  form,  it  appears  to 
derive  immense  probability  (to  say  the  least)  from  its 
simplicity.  Like  the  laws  of  motion,  when  stated  in  its 
most  general  form,  it  appears  to  carry  with  it  its  own 
evidence.  And  thus  this  great  theory  borrows  some 
thing  of  its  character  from  the  Ideas  which  it  involves, 
as  well  as  from  the  Experiments  by  which  it  was  esta 

8.  We  may  find  in  many  of  Mr.  Faraday's  subsequent 
reasonings,  clear  evidence  that  this  idea  of  the  connex 
ion  of  polarities,  as  now  developed,  is  not  limited  in  its 

*   Hist.  Iml.  Set.,  B.  xiv.  o.  ix.  sect  2.  +  Arts.  915,  916, 91?. 


application  to  facts  already  known  experimentally,  hut, 
like  other  ideas,  determines  the  philosopher's  researches 
into  the  unknown,  and  gives  us  the  form  of  knowledge 
even  before  we  possess  the  matter.  Thus,  he  says,  in 
his  Thirteenth  Series*,  "I  have  long  sought,  and  still 
seek,  for  an  effect  or  condition  which  shall  be  to  statical 
electricity  what  magnetic  force  is  to  current  electricity  ; 
for  as  the  lines  of  discharge  are  associated  with  a  cer 
tain  transverse  effect,  so  it  appeared  to  me  impossible 
but  that  the  lines  of  tension  or  of  inductive  action, 
which  of  necessity  precede  the  discharge,  should  also 
have  their  correspondent  transverse  condition  or  effect." 
Other  similar  passages  might  be  found. 

I  will  now  consider  another  case  to  which  we  may 
apply  the  principle  of  connected  polarities. 

9.  Connexion  of  Chemical  and  Crystalline  Polari 
ties. — The  close  connexion  between  the  chemical  affinity 
and  the  crystalline  attraction  of  elements  cannot  be 
overlooked.  Bodies  never  crystallize  but  when  their 
elements  combine  chemically ;  and  solid  bodies  which 
combine,  when  they  do  it  most  completely  and  exactly, 
also  crystallize.  The  forces  which  hold  together  the  ele 
ments  of  a  crystal  of  alum  are  the  same  forces  which 
make  it  a  crystal.  There  is  no  distinguishing  between 
the  two  sets  of  forces. 

Both  chemical  and  crystalline  forces  are  polar,  as  we 
stated  in  the  last  chapter;  but  the  polarity  in  the  two 
cases  is  of  a  different  kind.  The  polarity  of  chemical 
forces  is  then  put  in  the  most  distinct  form,  when  it  is 
identified  with  electrical  polarity ;  the  polarity  of  the 
particles  of  crystals  has  reference  to  their  geometrical 
form.  And  it  is  clear  that  these  two  kinds  of  polarity 
must  be  connected.  Accordingly,  Berzelius  expressly 
asserts t  the  necessary  identity  of  these  two  polarities. 

*   Art.  16f>8.  t   Essay  on  Chemical  Prop.,  113. 


"  The  regular  forms  of  'bodies  suppose  a  polarity  which 
can  be  no  other  than  an  electric  or  magnetic  polarity." 
This  being  so  seemingly  inevitable,  we  might  expect  to 
find  the  electric  forces  manifesting  some  relation  to  the 
definite  directions  of  crystalline  forms.  Mr.  Faraday 
tried,  but  in  vain,  to  detect  some  such  relation.  He 
attempted  to  ascertain*  whether  a  cube  of  rock  crystal 
transmitted  the  electrical  force  of  tension  with  different 
intensity  along  and  across  the  axis  of  the  crystal.  In 
the  first  specimen  there  seemed  to  be  some  difference ; 
but  in  other  experiments,  made  both  with  rock  crystal 
and  with  calc  spar,  this  difference  disappeared.  Al 
though  therefore  we  may  venture  to  assert  that  there 

O  *J 

must  be  some  very  close  connexion  between  electrical 
and  crystalline  forces,  we  are,  as  yet,  quite  ignorant 
what  the  nature  of  the  connexion  is,  and  in  what  kind 
of  phenomena  it  will  manifest  itself. 

10.  Connexion  of  Crystalline  and  Optical  Polarities. 
—Crystals  present  to  us  optical  phenomena  which  have 
a  manifestly  polar  character.  The  double  refraction, 
both  of  uniaxal  and  of  biaxal  crystals,  is  always  accom 
panied  with  opposite  polarization  of  the  two  rays ;  and 
in  this  and  in  other  ways  light  is  polarized  in  directions 
dependent  upon  the  axes  of  the  crystalline  form,  that  is, 
on  the  directions  of  the  polarities  of  the  crystalline  par 
ticles.  The  identity  of  these  two  kinds  of  polarity  (cry 
stalline  and  optical)  is  too  obvious  to  need  insisting  on  ; 
and  it  is  not  necessary  for  us  here  to  decide  by  what 
hypothesis  this  identity  may  most  properly  be  repre 
sented.  We  may  hereafter  perhaps  find  ourselves  jus 
tified  in  considering  the  crystalline  forces  as  determining 
the  elasticity  of  the  luminiferous  ether  to  be  different 
in  different  directions  within  the  crystal,  and  thus  as 
determining  the  refraction  and  polarization  of  the  light 

*    Rt'sean-Jics.     Art.  1689. 


which  the  crystal  transmits.  But  at  present  we  merely 
note  this  case  as  an  additional  example  of  the  manifest 
connexion  and  fundamental  identity  of  two  co-existent 

11.  Connexion  of  Polarities  in  general. — Thus  we 
find  that  the  connexion  of  different  kinds  of  polarities, 
magnetic,  electric,  chemical,  crystalline,  and  optical,  is 
certain  as  a  truth  of  experimental  science.  We  have 
attempted  to  show  further  that  in  the  minds  of  several 
of  the  most  eminent  discoverers  and  philosophers,  such 
a  conviction  is  something  more  than  a  mere  empirical 
result:  it  is  a  principle  which  has  regulated  their  re 
searches  while  it  was  still  but  obscurely  seen  and  imper 
fectly  unfolded,  and  has  given  to  their  theories  a  charac 
ter  of  generality  and  self-evidence  which  experience 
alone  cannot  bestow. 

It  will,  perhaps,  be  said  that  these  doctrines, — that 
scientific  researches  may  usefully  be  directed  by  prin 
ciples  in  themselves  vague  and  obscure ; — that  theories 
may  have  an  evidence  superior  to  and  anterior  to  expe 
rience  ; — are  doctrines  in  the  highest  degree  dangerous, 
and  utterly  at  variance  with  the  soundest  maxims  of 
modern  times  respecting  the  cultivation  of  science. 

To  the  justice  and  wisdom  of  this  caution  I  entirely 
agree :  and  although  I  have  shown  that  this  principle  of 
the  connexion  of  polarities,  rightly  interpreted  and  esta 
blished  in  each  case  by  experiment,  involves  profound 
and  comprehensive  truths ;  I  think  it  no  less  important 
to  remark  that,  at  least  in  the  present  stage  of  our 
knowledge,  we  can  make  no  use  of  this  principle  with 
out  taking  care,  at  every  step,  to  determine  by  clear  and 
decisive  experiments,  its  proper  meaning  and  applica 
tion.  All  endeavours  to  proceed  otherwise  have  led, 
and  must  lead,  to  ignorance  and  confusion.  Attempts 
to  deduce  from  our  bare  idea  of  polarity,  and  our  fun- 
VOL  i.  w.  r.  B  B 


damental  convictions  respecting  the  connexion  of  polari 
ties,  theories  concerning  the  forces  which  really  exist  in 
nature,  can  hardly  have  any  other  result  than  to  bewilder 
men's  minds,  and  to  misdirect  their  efforts. 

So  far,  indeed,  as  this  persuasion  of  a  connexion 
among  apparently  different  kinds  of  agencies,  impels 
men,  engaged  in  the  pursuit  of  knowledge,  to  collect 
observations,  to  multiply,  repeat,  and  vary  experiments, 
and  to  contemplate  the  result  of  these  in  all  aspects 
and  relations,  it  may  be  an  occasion  of  the  most  impor 
tant  discoveries.  Accordingly  we  find  that  the  great 
laws  of  phenomena  which  govern  the  motions  of  the 
planets  about  the  sun,  were  first  discovered  by  Kepler, 
in  consequence  of  his  scrutinizing  the  recorded  observa 
tions  with  an  intense  conviction  of  the  existence  of  geo 
metrical  and  arithmetical  harmonies  in  the  solar  system. 
Perhaps  we  may  consider  the  discovery  of  the  connexion 
of  magnetism  and  electricity  by  Professor  QErsted  in 
1820,  as  an  example  somewhat  of  the  same  kind;  for 
he  also  was  a  believer  in  certain  comprehensive  but  un 
defined  relations  among  the  properties  of  bodies ;  and 
in  consequence  of  such  views  entertained  great  admira 
tion  for  the  Prologue  to  the  Chemistry  of  the  Nineteenth 
Century,  of  Winterl,  already  mentioned.  M.  (Ersted,  in 
1803,  published  a  summary  of  this  work ;  and  in  so  do 
ing,  praised  the  views  of  Winterl  as  far  more  profound 
and  comprehensive  than  those  of  Lavoisier.  Soon  after 
wards  a  Review  of  this  publication  appeared  in  France  **, 
in  which  it  was  spoken  of  as  a  work  only  fit  for  the 
dark  ages,  and  as  the  indication  of  a  sect  which  had 
for  some  time  "  ravaged  Germany,"  and  inundated  that 
country  with  extravagant  and  unintelligible  mysticism. 
It  was,  therefore,  a  kind  of  triumph  to  M.  (Ersted  to 
bo,  after  some  years'  labour,  the  author  of  one  of  the 

*  Ann.  Chim.,  Tom.  t.  (1804),  p.  191. 


most  remarkable  and  fertile  physical  discoveries  of  his 

12.  It  was  not  indeed  without  some  reason  that  cer 
tain  of  the  German  philosophers  were  accused  of  dealing 
in  doctrines  vast  and  profound  in  their  aspect,  but,  in 
reality,  indefinite,  ambiguous,  and  inapplicable.  And 
the  most  prominent  of  such  doctrines  had  reference  to 
the  principle  now  under  our  consideration ;  they  repre 
sented  the  properties  of  bodies  as  consisting  in  certain 
polarities,  and  professed  to  deduce,  from  the  very  nature 
of  things,  with  little  or  no  reference  to  experiment,  the 
existence  and  connexion  of  these  polarities.  Thus  Schel- 
ling,  in  his  Ideas  towards  a  Philosophy  of  Nature,  pub 
lished  in  1803,  says*,  "Magnetism  is  the  universal  act 
of  investing  Multiplicity  with  Unity ;  but  the  universal 
form  of  the  reduction  of  Multiplicity  to  Unity  is  the 
Line,  pure  Longitudinal  Extension :  hence  Magnetism 
is  determination  of  pure  Longitudinal  Extension  ;  and 
as  this  manifests  itself  by  absolute  Cohesion,  Magnetism 
is  the  determination  of  absolute  Cohesion."  And  as 
Magnetism  was,  by  such  reasoning,  conceived  to  be 
proved  as  a  universal  property  of  matter,  Schelling  as 
serted  it  to  be  a  confirmation  of  his  views  when  it  was 
discovered  that  other  bodies  besides  iron  are  magnetic. 
In  like  manner  he  used  such  expressions  as  the  follow- 
ingf:  "The  threefold  character  of  the  Universal,  the 
Particular,  and  the  Indifference  of  the  two, — as  ex 
pressed  in  their  Identity,  is  Magnetism,  as  expressed 
in  their  Difference,  is  Electricity,  and  as  expressed  in 
the  Totality,  is  Chemical  Process.  Thus  these  forms 
are  only  one  form ;  and  the  Chemical  Process  is  a  mere 
transfer  of  the  three  Points  of  Magnetism  into  the  Tri 
angle  of  Chemistry." 

It  was  very  natural  that  the  chemists  should  refuse 
*   P.  2'2'A.  +   P.  -J8<). 

It  IJ '-' 


to  acknowledge,  in  this  fanciful  and  vague  language, 
(delivered,  however,  it  is  to  be  recollected,  in  1803,)  an 
anticipation  of  Davy's  doctrine  of  the  identity  of  electri 
cal  and  chemical  forces,  or  of  (Ersted's  electro-magnetic 
agency.  Yet  it  was  perhaps  no  less  natural  that  the 
author  of  such  assertions  should  look  upon  every  great 
step  in  the  electro-chemical  theory  as  an  illustration 
of  his  own  doctrines.  Accordingly  we  find  Schelling 
welcoming,  with  a  due  sense  of  their  importance,  the 
discoveries  of  Faraday.  When  he  heard  of  the  experi 
ment  in  which  electricity  was  produced  from  common 
magnetism,  he  fastened  with  enthusiasm  upon  the  dis 
covery,  even  before  he  knew  any  of  its  details,  and  pro 
claimed  it  at  a  public  meeting  of  a  scientific  body*  as 
one  of  the  most  important  advances  of  modern  science. 
We  have  (he  thus  reasoned)  three  effects  of  polar  forces ; 
— electro-chemical  Decomposition,  electrical  Action, 
Magnetism.  Volta  and  Davy  had  confirmed  experimen 
tally  the  identity  of  the  two  former  agencies :  (Ersted 
showed  that  a  closed  voltaic  circuit  acquired  magnetic 
properties :  but  in  order  to  exhibit  the  identity  of  elec 
tric  and  magnetic  action  it  was  requisite  that  electric 
forces  should  be  extricated  from  magnetic.  This  great 
step  Faraday,  he  remarked,  had  made,  in  producing  the 
electric  spark  by  means  of  magnets. 

13.  Although  conjectures  and  assertions  of  the  kind 
thus  put  forth  by  Schelling  involve  a  persuasion  of  the 
pervading  influence  and  connexion  of  polarities,  which 
persuasion  has  already  been  confirmed  in  many  instances, 
they  involve  this  principle  in  a  manner  so  vague  and 
ambiguous  that  it  can  rarely,  in  such  a  form,  be  of 
any  use  or  value.  Such  views  of  polarity  can  never 
teach  us  in  what  cases  we  are  and  in  what  we  are  not 
to  expect  to  find  polar  relations ;  and  indeed  tend  rather 

*  Ucbor  Faradav's  Ncnesle  Enldeckiins.      Miinchen.   1832. 


to  diffuse  error  and  confusion,  than  to  promote  know 
ledge.  Accordingly  we  cannot  be  surprized  to  find  such 
doctrines  put  forward  by  their  authors  as  an  evidence  of 
the  small  value  and  small  necessity  of  experimental 
science.  This  is  done  by  the  celebrated  metaphysician 
Hegel,  in  his  Encyclopaedia*.  '"Since,"  says  he,  "the 
plane  of  incidence  and  of  reflection  in  simple  reflection 
is  the  same  plane,  when  a  second  reflector  is  introduced 
which  further  distributes  the  illumination  reflected  from 
the  first,  the  position  of  the  first  plane  with  respect  to 
the  second  plane,  containing  the  direction  of  the  first 
reflection  and  of  the  second,  has  its  influence  upon  the 
position,  illumination  or  darkening  of  the  object  as  it 
appears  by  the  second  reflection.  This  influence  must 
be  the  strongest  when  the  two  planes  are  what  we  must 
call  negatively  related  to  each  other: — that  is,  when 
they  are  at  right  angles."  "  But,"  he  adds,  "  when  men 
infer  (as  Mai  us  has  done)  from  the  modification  which 
is  produced  by  this  situation,  in  the  illumination  of  the 
reflection,  that  the  molecules  of  light  in  themselves, 
that  is,  on  their  different  sides,  possess  different  physical 
energies ;  and  when  on  this  foundation,  along  with  the 
phenomena  of  entoptical  colours  therewith  connected,  a 
wide  labyrinth  of  the  most  complex  theory  is  erected ; 
we  have  then  one  of  the  most  remarkable  examples  of 
the  inferences  of  physics  from  experiment."  If  Hegel's 
reasoning  prove  anything,  it  must  prove  that  polariza 
tion  always  accompanies  reflection  under  such  circum 
stances  as  he  describes :  yet  all  physical  philosophers 
know  that  in  the  case  of  metals,  in  which  the  reflection 
is  most  complete,  light  is  not  completely  polarized  at 
any  angle ;  and  that  in  other  substances  the  polarization 
depends  upon  various  circumstances  which  show  how 
idle  and  inapplicable  is  the  account  he  thus  gives  of  the 

*  Sec.  278. 


property.  His  self-complacent  remark  about  the  infer 
ences  of  physics  from  experiment,  is  intended  to  recom 
mend  by  comparison  his  own  method  of  considering  the 
nature  of  things  in  themselves ;  a  mode  of  obtaining 
physical  truth  which  had  been  more  than  exhausted  by 
Aristotle,  and  out  of  which  no  new  attempts  have  ex 
tracted  anything  of  value  since  his  time. 

14.  Thus  the  general  conclusion  to  which  we  are  led 
on  this  subject  is,  that  the  persuasion  of  the  existence 
and  connexion  or  identity  of  various  polarities  in  nature, 
although  very  naturally  admitted,  and  in  many  cases 
interpreted  and  confirmed  by  observed  facts,  is  of  itself, 
so  far  as  we  at  present  possess  it,  a  very  insecure  guide 
to  scientific  doctrines.  When  it  is  allowed  to  dictate 
our  theories,  instead  of  animating  and  extending  our 
experimental  researches,  it  leads  only  to  errour,  confusion, 
obscurity,  and  mysticism. 

This  Fifth  Book,  on  the  subject  of  Polarities,  is  a 
short  one  compared  with  most  of  the  others.  This 
arises  in  a  great  measure  from  the  circumstance  that  the 
Idea  of  Polarity  has  only  recently  been  apprehended  and 
applied,  with  any  great  degree  of  clearness,  among  phy 
sical  philosophers ;  and  is  even  yet  probably  entertained 
in  an  obscure  and  ambiguous  manner  by  most  experi 
mental  inquirers.  I  have  been  desirous  of  not  attempt 
ing  to  bring  forward  any  doctrines  upon  the  subject, 
except  such  as  have  been  fully  illustrated  and  exemplified 
by  the  acknowledged  progress  of  the  physical  sciences. 
If  I  had  been  willing  to  discuss  the  various  speculations 
which  have  been  published  respecting  the  universal  pre 
valence  of  polarities  in  the  universe,  and  their  results  in 
every  province  of  nature,  I  might  easily  have  presented 
this  subject  in  a  more  extended  form ;  but  this  would 
not  have  been  consistent  with  my  plan  of  tracing  the 
influence  of  scientific  ideas  only  so  far  as  they  have  really 


aided  in  disclosing  and  developing  scientific  truths.  And 
as  the  influence  of  this  idea  is  clearly  distinguishable 
both  from  those  which  precede  and  those  which  follow  in 
the  character  of  the  sciences  to  which  it  gives  rise,  and 
appears  likely  to  be  hereafter  of  great  extent  and  conse 
quence,  it  seemed  better  to  treat  of  it  in  a  separate 
Book,  although  of  a  brevity  disproportioned  to  the 






1.  WE  have  now  to  bring  into  view,  if  possible,  the 
ideas  and  general  principles  which  are  involved  in  Che 
mistry, — the  science  of  the  composition  of  bodies.  For  in 
this  as  in  other  parts  of  human  knowledge,  we  shall  find 
that  there  are  certain  ideas,  deeply  seated  in  the  mind, 
though  shaped  and  unfolded  by  external  observation, 
which  are  necessary  conditions  of  the  existence  of  such 
a  science.  These  ideas  it  is,  which  impel  man  to  such 
a  knowledge  of  the  composition  of  bodies,  which  give 
meaning  to  facts  exhibiting  this  composition,  and  uni 
versality  to  special  truths  discovered  by  experience. 
These  are  the  Ideas  of  Element  and  of  Substance. 

Unlike  the  idea  of  polarity,  of  which  we  treated  in 
the  last  Book,  these  ideas  have  been  current  in  men's 
minds  from  very  early  times,  and  formed  the  subject  of 
some  of  the  first  speculations  of  philosophers.  It  hap 
pened  however,  as  might  have  been  expected,  that  in  the 
first  attempts  they  were  not  clearly  distinguished  from 
other  notions,  and  were  apprehended  and  applied  in  an 
obscure  and  confused  manner.  We  cannot  better  ex 
hibit  the  peculiar  character  and  meaning  of  these  ideas 
than  by  tracing  the  form  which  they  have  assumed  and 


the  efficacy  which  they  have  exerted  in  these  successive 
essays.  This,  therefore,  I  shall  endeavour  to  do,  begin 
ning  with  the  Idea  of  Element. 

2.  That  bodies  are  composed  or  made  up  of  certain 
parts,  elements,  or  principles,  is  a  conception  which  has 
existed  in  men's  minds  from  the  beginning  of  the  first 
attempts  at  speculative  knowledge.  The  doctrine  of  the 
Four  Elements,  earth,  air,  fire  and  water,  of  which  all 
things  in  the  universe  were  supposed  to  be  constituted, 
is  one  of  the  earliest  forms  in  which  this  conception  was 
systematized ;  and  this  doctrine  is  stated  by  various 
authors  to  have  existed  as  early  as  the  times  of  the 
ancient  Egyptians"''.  The  words  usually  employed  by 
Greek  writers  to  express  these  elements  are  dpxh  a  prin 
ciple  or  beginning,  and  aToi-^elov,  which  probably  meant 
a  letter  (of  a  word)  before  it  meant  an  element  of  a 
compound.  For  the  resolution  of  a  word  into  its  letters 
is  undoubtedly  a  remarkable  instance  of  a  successful 
analysis  performed  at  an  early  stage  of  man's  history ; 
and  might  very  naturally  supply  a  metaphor  to  denote 
the  analysis  of  substances  into  their  intimate  parts,  when 
men  began  to  contemplate  such  an  analysis  as  a  subject 
of  speculation.  The  Latin  word  elementum  itself,  though 
by  its  form  it  appears  to  be  a  derivative  abstract  term, 
comes  from  some  root  now  obsolete ;  probably  f  from  a 
word  signifying  to  grow  or  spring  up. 

The  mode  in  which  elements  form  the  compound 
bodies  and  determine  their  properties  was  at  first,  as 
might  be  expected,  vaguely  and  variously  conceived.  It 
will,  I  trust,  hereafter  be  made  clear  to  the  reader  that 

*  Gilbert's  Phys.,  L.  i.  c.iii. 

t  Vossius  in  voce.  "  Conjccto  esse  ab  antiqua  voco  eleo  pro  oleo, 
id  est  cresco :  a  qua  significationc  proles,  suboles,  adolescent :  ut  ab 
juratum,  juramentum ;  ab  adjitlum,  adjnmcnhnn :  sic  ab  delum, 
elcmenhnn  :  quia  intle  oninia  crcscunt  ac  nascuntur." 

378  PHILOSOPHY    OF    C11EMIST11Y. 

the  relation  of  the  elements  to  the  compound  involves  a 
peculiar  and  appropriate  Fundamental  Idea,  not  suscept 
ible  of  being  correctly  represented  by  any  comparison  or 
combination  of  other  ideas,  and  guiding  us  to  clear  and 
definite  results  only  when  it  is  illustrated  and  nourished 
by  an  abundant  supply  of  experimental  facts.  But  at  first 
the  peculiar  and  special  notion  which  is  required  in  a  just 
conception  of  the  constitution  of  bodies  was  neither  dis 
cerned  nor  suspected ;  and  up  to  a  very  late  period  in  the 
history  of  chemistry,  men  went  on  attempting  to  appre 
hend  the  constitution  of  bodies  more  clearly  by  substi 
tuting  for  this  obscure  and  recondite  idea  of  Elementary 
Composition,  some  other  idea  more  obvious,  more  lumi 
nous,  and  more  familiar,  such  as  the  ideas  of  Resem 
blance,  Position,  and  mechanical  Force.  We  shall  briefly 
speak  of  some  of  these  attempts,  and  of  the  errours  which 
were  thus  introduced  into  speculations  on  the  relations 
of  elements  and  compounds. 

3.  Compounds  assumed  to  resemble  their  Elements. — 
The  first  notion  was  that  compounds  derive  their  quali 
ties  from  their  elements  by  resemblance : — they  are  hot 
in  virtue  of  a  hot  element,  heavy  in  virtue  of  a  heavy 
element,  and  so  on.  In  this  way  the  doctrine  of  the  four 
elements  was  framed ;  for  every  body  is  either  hot  or 
cold,  moist  or  dry ;  and  by  combining  these  qualities  in 
all  possible  ways,  men  devised  four  elementary  sub 
stances,  as  has  been  stated  in  the  History"". 

This  assumption  of  the  derivation  of  the  qualities  of 
bodies  from  similar  qualities  in  the  elements  was,  as  we 
shall  see,  altogether  baseless  and  unphilosophical,  yet  it 
prevailed  long  and  universally.  It  was  the  foundation  of 
medicine  for  a  long  period,  both  in  Europe  and  Asia; 
disorders  being  divided  into  hot,  cold,  and  the  like ;  and 
remedies  being  arranged  according  to  similar  distinctions. 

*   Hist.  Ind  Sci.,  B.  i.  c.  ii.  sect.  2. 


Many  readers  will  recollect,  perhaps,  the  story*  of  the 
indignation  which  the  Persian  physicians  felt  towards  the 
European,  when  he  undertook  to  cure  the  ill  effects  of 
cucumber  upon  the  patient,  by  means  of  mercurial  medi 
cine  :  for  cucumber,  which  is  cold,  could  not  be  coun 
teracted,  they  maintained,  by  mercury,  which  in  their 
classification  is  cold  also.  Similar  views  of  the  operation 
of  medicines  might  easily  be  traced  in  our  own  country. 
A  moment's  reflection  may  convince  us  that  when  drugs 
of  any  kind  are  subjected  to  the  chemistry  of  the 
human  stomach  and  thus  made  to  operate  on  the  human 
frame,  it  is  utterly  impossible  to  form  the  most  remote 
conjecture  what  the  result  will  be  from  any  such  vague 
notions  of  their  qualities  as  the  common  use  of  our 
senses  can  give.  And  in  like  manner  the  common  ope 
rations  of  chemistry  give  rise  in  almost  every  instance 
to  products  which  bear  no  resemblance  to  the  materials 
employed.  The  results  of  the  furnace,  the  alembic,  the 
mixture,  frequently  have  no  visible  likeness  to  the 
ingredients  operated  upon.  Iron  becomes  steel  by  the 
addition  of  a  little  charcoal ;  but  what  visible  trace  of 
the  charcoal  is  presented  by  the  metal  thus  modified  ? 
The  most  beautiful  colours  are  given  to  glass  and 
earthenware  by  minute  portions  of  the  ores  of  black  or 
dimjv  metals,  as  iron  and  manganese.  The  worker  in 


metal,  the  painter,  the  dyer,  the  vintner,  the  brewer, 
all  the  artisans  in  short  who  deal  with  practical  che 
mistry,  are  able  to  teach  the  speculative  chemist  that 
it  is  an  utter  mistake  to  expect  that  the  qualities  of  the 
elements  shall  be  still  discoverable,  in  an  unaltered  form, 
in  the  compound.  This  first  rude  notion  of  an  element, 
that  it  determines  the  properties  of  bodies  by  resem 
blance,  must  be  utterly  rejected  and  abandoned  before 

*  See  Hadji  Baba. 


we  can  make  any  advance  towards  a  true  apprehension 
of  the  constitution  of  bodies. 

4.  This  step  accordingly  was  made,  when  the  hypo 
thesis  of  the  four  elements  was  given  up,  and  the  doc 
trine  of  the  three  Principles,  Salt,  Sulphur  and  Mercury, 
was  substituted  in  its  place.    For  in  making  this  change, 
as  I  have  remarked  in  the  History*,  the  real  advance 
was  the  acknowledgment  of  the  changes  produced  by 
the  chemist's  operations  as  results  to  be  accounted  for 
by  the  union  and  separation   of  substantial    elements, 
however  great    the   changes,  and    however  unlike  the 
product  might  be  to  the  materials.     And  this  step  once 
made,  chemists  wrent  on  constantly  advancing  towards 
a  truer  view  of  the  nature  of  an  element,  and  conse 
quently,  towards  a  more  satisfactory  theory  of  chemical 

5.  Yet  we  may,  I  think,  note  one  instance,  even  in 
the  works  of  eminent  modern  chemists,  in  which  this 
maxim,   that  we  have  no  right  to  expect  any  resem 
blance  between  the  elements  and  the  compound,  is  lost 
sight  of.     I  speak  of  certain  classifications  of  mineral 
substances.    Berzelius,  in  his  System  of  Mineral  Arrange 
ment,  places  sulphur  next  to  the  sulphur ets.    But  surely 
this  is  an  errour,  involving  the  ancient  assumption  of 
the  resemblance  of  elements  and  compounds ;  as  if  we 
were  to  expect  the  sulphurets  to  bear  a  resemblance  to 
sulphur.     All  classifications  are  intended  to  bring  toge 
ther  things  resembling  each  other :   the  sulphurets  of 
metals  have  certain  general  resemblances  to  each  other 
which  make  them  a  tolerably  distinct,  well  determined, 
class  of  bodies.     But  sulphur  has  no  resemblances  with 
these,  and  no  analogies  with  them,  either  in  physical 
or  even  in  chemical  properties.     It  is  a  simple  body; 

*   Hist,  hid.  Sci.,  B.  iv.  {.:  i. 


and  both  its  resemblances  and  its  analogies  direct  us  to 
place  it  along  with  other  simple  bodies,  (selenium,  and 
phosphorus,)  which,  united  with  metals,  produce  com 
pounds  not  very  different  from  the  sulphurets.  Sulphur 
cannot  be,  nor  approach  to  being,  a  sulphuret ;  we  must 
not  confound  what  it  is  with  what  it  makes.  Sulphur 
has  its  proper  influence  in  determining  the  properties  of 
the  compound  into  which  it  enters ;  but  it  does  not  do 
this  according  to  resemblance  of  qualities,  or  according 
to  any  principle  which  properly  leads  to  propinquity  in 
classification.  • 

6.  Compounds  assumed  to  be  determined  by  the  Figure 
of  Elements. — I  pass  over  the  fanciful  modes  of  represent 
ing  chemical  changes  which  were  employed  by  the  Alche 
mists  ;  for  these  strange  inventions  did  little  in  leading 
men  towards  a  juster  view  of  the  relations  of  elements  to 
compounds.  I  proceed  for  an  instant  to  the  attempt  to 
substitute  another  obvious  conception  for  the  still  obscure 
notion  of  elementary  composition.  It  was  imagined  that 
all  the  properties  of  bodies  and  their  mutual  operations 
might  be  accounted  for  by  supposing  them  constituted  of 
particles  of  various  forms,  round  or  angular,  pointed  or 
hooked,  straight  or  spiral.  This  is  a  very  ancient  hypo 
thesis,  and  a  favourite  one  with  many  casual  speculators 
in  all  ages.  Thus  Lucretius  undertakes  to  explain  why 
wine  passes  rapidly  through  a  sieve  and  oil  slowly,  by 
telling  us  that  the  latter  substance  has  its  particles  either 
larger  than  those  of  the  other,  or  more  hooked  and  inter 
woven  together.  And  he  accounts  for  the  difference  of 
sweet  and  bitter  by  supposing  the  particles  in  the  former 
case  to  be  round  and  smooth,  in  the  latter  sharp  and 
jagged*.  Similar  assumptions  prevailed  in  modern  times 
on  the  revival  of  the  mechanical  philosophy,  and  consti 
tute  a  large  part  of  the  physical  schemes  of  Descartes 

*   De  Rrrttm  Nalura,  n.  390  siqq. 


and  Gassendi.  They  were  also  adopted  to  a  considerable 
extent  by  the  chemists.  Acids  were  without  hesitation 
assumed  to  consist  of  sharp  pointed  particles ;  which,  "  I 
hope,"  Lemery  says  *,  "  no  one  will  dispute,  seeing  every 
one's  experience  does  demonstrate  it :  he  needs  but  taste 
an  acid  to  be  satisfied  of  it,  for  it  pricks  the  tongue  like 
anything  keen  and  finely  cut."  Such  an  assumption  is 
not  only  altogether  gratuitous  and  useless,  but  appears  to 
be  founded  in  some  degree  upon  a  confusion  in  the  meta 
phorical  and  literal  use  of  such  words  as  keen  and  sharp. 
The  assumption  once  made,  it  was  easy  to  accommodate 
it,  in  a  manner  equally  arbitrary,  to  other  facts.  "A 
demonstrative  and  convincing  proof  that  an  acid  does 
consist  of  pointed  parts  is,  that  not  only  all  acid  salts  do 
crystallize  into  edges,  but  all  dissolutions  of  different 
things,  caused  by  acid  liquors,  do  assume  this  figure  in 
their  crystallization.  These  crystals-  consist  of  points 
differing  both  in  length  and  bigness  one  from  another, 
and  this  diversity  must  be  attributed  to  the  keener  or 
blunter  edges  of  the  different  sorts  of  acids  :  and  so  like 
wise  this  difference  of  the  points  in  subtilty  is  the  cause 
that  one  acid  can  penetrate  and  dissolve  with  one  sort  of 
mixt,  that  another  can't  rarify  at  all :  Thus  vinegar  dis 
solves  lead,  which  aquafortis  can't :  aquafortis  dissolves 
quicksilver,  which  vinegar  will  not  touch ;  aqua  regalis 
dissolves  gold,  whenas  aquafortis  cannot  meddle  with  it ; 
on  the  contrary,  aqua  fortis  dissolves  silver,  but  can  do 
nothing  with  gold,  and  so  of  the  rest." 

The  leading  fact  of  the  vehement  combination  and 
complete  union  of  acid  and  alkali  readily  suggested  a  fit 
form  for  the  particles  of  the  latter  class  of  substances. 
"  This  effect,"  Lemery  adds,  "  may  make  us  reasonably 
conjecture  that  an  alkali  is  a  terrestrious  and  solid  mat 
ter  whose  forms  are  figured  after  such  a  manner  that  the 
*  Chemistry,  p.  25. 


acid  points  entering  in  do  strike  and  divide  whatever 
opposes  their  motion."  And  in  a  like  spirit  are  the  spe 
culations  in  Dr.  Mead's  Mechanical  Account  of  Poisons 
(1745).  Thus  he  explains  the  poisonous  effect  of  corro 
sive  sublimate  of  mercury  by  saying*  that  the  particles  of 
the  salt  are  a  kind  of  lamellae  or  blades  to  which  the 
mercury  gives  an  additional  weight.  If  resublimed  with 
three-fourths  the  quantity  of  mercury,  it  loses  its  corro- 
siveness,  (becoming  calomel,)  which  arises  from  this,  that 
in  sublimation  "  the  crystalline  blades  are  divided  every 
time  more  and  more  by  the  force  of  the  fire ;"  and  "  the 
broken  pieces  of  the  crystals  uniting  into  little  masses  of 
differing  figures  from  their  former  make,  those  cutting 
points  are  now  so  much  smaller  that  they  cannot  make 
wounds  deep  enough  to  be  equally  mischievous  and 
deadly :  and  therefore  do  only  vellicate  and  twitch  the 
sensible  membranes  of  the  stomach." 

7.  Among  all  this  very  fanciful  and  gratuitous  assump 
tion  we  may  notice  one  true  principle  clearly  introduced, 
namely,  that  the  suppositions  which  we  make  respecting 
the  forms  of  the  elementary  particles  of  bodies  and  their 
mode  of  combination  must  be  such  as  to  explain  the  facts 
of  crystallization,  as  well  as  of  mere  chemical  change. 
This  principle  we  shall  hereafter  have  occasion  to  insist 
upon  further. 

I  now  proceed  to  consider  a  more  refined  form  of 
assumption  respecting  the  constitution  of  bodies,  yet  still 
one  in  which  a  vain  attempt  is  made  to  substitute  for  the 
peculiar  idea  of  chemical  composition  a  more  familiar 
mechanical  conception. 

8.  Compounds  assumed  to  be  determined  by  the  Mecha 
nical  Attraction  of  the  Elements. — When,  in  consequence 
of  the  investigations  and  discoveries  of  Newton  and  his 
predecessors,  the  conception   of  mechanical  force  had 

*  P.  H>9. 


become  clear  and  familiar,  so  far  as  the  action  of  exter 
nal  forces  upon  a  body  was  concerned,  it  was  very  natural 
that  the  mathematicians  who  had  pursued  this  train  of 
speculation  should  attempt  to  apply  the  same  conception 
to  that  mutual  action  of  the  internal  parts  of  a  body  by 
which  they  are  held  together.  Newton  himself  had 
pointed  the  way  to  this  attempt,  In  the  Preface  to  the 
Principia,  after  speaking  of  what  he  has  done  in  calcu 
lating  the  effects  of  forces  upon  the  planets,  satellites, 
&c.,  he  adds,  "  Would  it  were  permitted  us  to  deduce  the 
other  phenomena  of  nature  from  mechanical  principles 
by  the  same  kind  of  reasoning.  For  many  things  move 
me  to  suspect  that  all  these  phenomena  depend  upon 
certain  forces,  by  which  the  particles  of  bodies,  through 
causes  not  yet  known,  are  either  urged  towards  each 
other,  and  cohere  according  to  regular  figures,  or  are 
repelled  and  recede  from  each  other ;  which  forces  being 
unknown,  philosophers  have  hitherto  made  their  attempts 
upon  nature  in  vain."  The  same  thought  is  at  a  later 
period  followed  out  further  in  one  of  the  Queries  at  the 
end  of  the  Opticks*.  "Have  not  the  small  particles  of 
bodies  certain  Powers,  Virtues,  or  Forces,  by  which  they 
act  at  a  distance,  not  only  upon  the  rays  of  light  for 
reflecting,  refracting  and  inflecting  them,  but  also  upon 
one  another  for  producing  a  great  part  of  the  phenomena 
of  nature  ?"  And  a  little  further  on  he  proceeds  to 
apply  this  expressly  to  chemical  changes.  "  When  Salt 
of  Tartar  runs  per  deliquium  [or  as  we  now  express  it, 
deliquesces]  is  not  this  done  by  an  attraction  between 
the  particles  of  the  Salt  of  Tartar  and  the  particles  of 
the  water  which  float  in  the  air  in  the  form  of  vapours  ? 
And  why  does  not  common  salt,  or  saltpetre,  or  vitriol, 
run  per  deliquium,  but  for  want  of  such  an  attraction  ?  or 
why  does  not  Salt  of  Tartar  draw  more  water  out  of  the 

"    Query  31 . 


air  than  in  a  certain  proportion  to  its  quantity,  but  for 
want  of  an  attractive  force  after  it  is  saturated  with 
water?"  He  goes  on  to  put  a  great  number  of  similar 
cases,  all  tending  to  the  same  point,  that  chemical  com 
binations  cannot  be  conceived  in  any  other  way  than  as 
an  attraction  of  particles. 

9.  Succeeding  speculators  in  his  school  attempted  to 
follow  out  this  view.  Dr.  Frend,  of  Christ  Church,  in 
1710,  published  his  Prwlectiones  Chymicce,  in  quibus 
omnes  fere  Operationcs  Chymicce  ad  xera  PrincApia 
ex  ipsius  Naturce  Lcc/ilus  rcdiyuntur.  Oxonii  liabitw. 
This  book  is  dedicated  to  Newton,  and  in  the  dedication, 
the  promise  of  advantage  to  chemistry  from  the  influence 
of  the  Newtonian  discoveries  is  spoken  of  somewhat 
largely, — much  more  largely,  indeed,  than  has  yet  been 
justified  by  the  sequel.  After  declaring  in  strong  terms 
that  the  only  prospect  of  improving  science  consists  in 
following  the  footsteps  of  Newton,  the  author  adds, 
"  That  force  of  attraction,  of  which  you  first  so  success 
fully  traced  the  influence  in  the  heavenly  bodies,  ope 
rates  in  the  most  minute  corpuscles,  as  you  long  ago 
hinted  in  your  Principia,  and  have  lately  plainly  shown 
in  your  Opticks ;  and  this  force  we  are  only  just  begin 
ning  to  perceive  and  to  study.  Under  these  circum 
stances  I  have  been  desirous  of  trying  what  is  the  result 
of  this  view  in  chemistry."  The  work  opens  formally 
enough,  with  a  statement  of  general  mechanical  prin 
ciples,  of  which  the  most  peculiar  are  these : — That 
there  exists  an  attractive  force  by  which  particles  when 
at  very  small  distances  from  each  other,  are  drawn  to 
gether; — that  this  force  is  different,  according  to  the 
different  figure  and  density  of  the  particles ; — that  the 
force  may  be  greater  on  one  side  of  a  particle  than  on 
the  other; — that  the  force  by  which  particles  cohere 
together  arises  from  attraction,  and  is  variously  modi- 
VOL.  i.  \v.  P.  C  c 


fied  according  to  the  quantity  of  contacts."  But  these 
principles  are  not  applied  in  any  definite  manner  to  the 
explanation  of  specific  phenomena.  He  attempts,  in 
deed,  the  question  of  special  solvents*.  Why  does  aqua 
fortis  dissolve  silver  and  not  gold,  while  aqua  regia 
dissolves  gold  and  not  silver?  which,  he  says,  is  the 
most  difficult  question  in  chemistry,  and  which  is  cer 
tainly  a  fundamental  question  in  the  formation  of  che 
mical  theory.  He  solves  it  by  certain  assumptions 
respecting  the  forces  of  attraction  of  the  particles,  and 
also  the  diameter  of  the  particles  of  the  acids  and  the 
pores  of  the  metals,  all  which  suppositions  are  gratuitous. 

10.  We  may  observe  further,  that  by  speaking,  as  I 
have  stated  that  he  does,  of  the  figure  of  particles,  he 
mixes  together  the  assumption  of  the  last  section  with 
the  one  which  we  are  considering  in  this.  This  com 
bination  is  very  unphilosophical,  or,  to  say  the  least, 
very  insufficient,  since  it  makes  a  new  hypothesis  neces 
sary.  If  a  body  be  composed  of  cubical  particles,  held 
together  by  their  mutual  attraction,  by  what  force  are 
the  parts  of  each  cube  held  together  ?  In  order  to  un 
derstand  their  structure,  we  are  obliged  again  to  assume 
a  cohesive  force  of  the  second  order,  binding  together 
the  particles  of  each  particle.  And  therefore  Newton 
himself  says  f,  very  justly,  "The  parts  of  all  homogeneal 
hard  bodies  which  fully  touch  each  other,  stick  together 
very  strongly :  and  for  explaning  how  this  is,  some  have 
invented  hooked  atoms,  which  is  legging  the  question" 
For  (he  means  to  imply,)  how  do  the  parts  of  the  hook 
stick  together? 

The  same  remark  is  applicable  to  all  hypotheses  in 

which  particles  of  a  complex  structure  are  assumed  as 

the  constituents  of  bodies :  for  while  we  suppose  bodies 

and  their  known  properties  to  result  from  the  mutual 

*  P.  54.  t  Oplich;  p.  304. 


actions  of  these  particles,  we  are  compelled  to  suppose 
the  parts  of  each  particle  to  be  held  together  by  forces 
still  more  difficult  to  conceive,  since  they  are  disclosed 
only  by  the  properties  of  these  particles,  which  as  yet 
are  unknown.  Yet  Newton  himself  has  not  abstained 
from  such  hypotheses :  thus  he  says  *,  "  A  particle  of  a 
salt  may  be  compared  to  a  chaos,  being  dense,  hard,  dry, 
and  earthy  in  the  center,  and  moist  and  watery  in  the 

Since  Newton's  time  the  use  of  the  term  attraction, 
as  expressing  the  cause  of  the  union  of  the  chemical 
elements  of  bodies,  has  been  familiarly  continued ;  and 
has,  no  doubt,  been  accompanied  in  the  minds  of  many 
persons  with  an  obscure  notion  that  chemical  attraction 
is,  in  some  way,  a  kind  of  mechanical  attraction  of  the 
particles  of  bodies.  Yet  the  doctrine  that  chemical  "  at 
traction"  and  mechanical  attraction  are  forces  of  the 
same  kind  has  never,  so  far  as  I  am  aware,  been  worked 
out  into  a  system  of  chemical  theory ;  nor  even  applied 
with  any  distinctness  as  an  explanation  of  any  particular 
chemical  phenomena.  Any  such  attenpt,  indeed,  could 
only  tend  to  bring  more  clearly  into  view  the  entire 
inadequacy  of  such  a  mode  of  explanation.  For  the 
leading  phenomena  of  chemistry  are  all  of  such  a  nature 
that  no  mechanical  combination  can  serve  to  express 
them,  without  an  immense  accumulation  of  additional 
hypotheses.  If  we  take  as  our  problem  the  changes  of 
colour,  transparency,  texture,  taste,  odour,  produced  by 
small  changes  in  the  ingredients,  how  can  we  expect  to 
give  a  mechanical  account  of  these,  till  we  can  give 
a  mechanical  account  of  colour,  transparency,  texture, 
taste,  odour,  themselves  ?  And  if  our  mechanical  hypo 
thesis  of  the  elementary  constitution  of  bodies  does  not 
explain  such  phenomena  as  those  changes,  what  can  it 

*  Oplicks,  p.  3G2. 


388  miLosoriiY  or  CHEMISTRY. 

explain,  or  what  can  be  the  value  of  it  ?  I  do  not  here 
insist  upon  a  remark  which  will  afterwards  come  before 
us,  that  even  crystalline  form,  a  phenomenon  of  a  far 
more  obviously  mechanical  nature  than  those  just  al 
luded  to,  has  never  yet  been  in  any  degree  explained  by 
such  assumptions  as  this,  that  bodies  consist  of  elemen 
tary  particles  exerting  forces  of  the  same  nature  as  the 
central  forces  which  we  contemplate  in  Mechanics. 

When  therefore  Newton  asks,  "  When  some  stones, 
as  spar  of  lead,  dissolved  in  proper  menstruums,  become 
salts,  do  not  these  things  show  that  salts  are  dry  earth 
and  watery  acid  united  by  attraction  f  we  may  answer, 
that  this  mode  of  expression  appears  to  be  intended  to 
identify  chemical  combination  with  mechanical  attrac 
tion  ; — that  there  would  be  no  objection  to  any  such 
identification,  if  we  could,  in  that  way,  explain,  or  even 
classify  well,  a  collection  of  chemical  facts ;  but  that 
this  has  never  yet  been  done  by  the  help  of  such  expres 
sions.  Till  some  advance  of  this  kind  can  be  pointed 
out,  we  must  necessarily  consider  the  power  which  pro 
duces  chemical  combination  as  a  peculiar  principle,  a 
special  relation  of  the  elements,  not  rightly  expressed  in 
mechanical  terms.  And  we  now  proceed  to  consider 
this  relation  under  the  name  by  which  it  is  most  fami 
liarly  known. 



1.  THE  earlier  chemists  did  not  commonly  involve 
themselves  in  the  confusion  into  which  the  mechanical 
philosophers  ran,  of  comparing  chemical  to  mechanical 
forces.  Their  attention  was  engaged,  and  their  ideas 


were  moulded,  by  their  own  pursuits.  They  saw  that 
the  connexion  of  elements  and  compounds  with  which 
they  had  to  deal,  was  a  peculiar  relation  which  must  be 
studied  directly ;  and  which  must  be  understood,  if  un 
derstood  at  all,  in  itself,  and  not  by  comparison  with  a 
different  class  of  relations.  At  different  periods  of  the 
progress  of  chemistry,  the  conception  of  this  relation, 
still  vague  and  obscure,  was  expressed  in  various  man 
ners;  and  at  last  this  conception  was  clothed  in  tole 
rably  consistent  phraseology,  and  the  principles  which  it 
involved  were,  by  the  united  force  of  thought  and  expe 
riment,  brought  into  view. 

2.  The  power  by  which  the  elements  of  bodies  com 
bine  chemically,  being,  as  we  have  seen,  a  peculiar  agency, 
different  from  mere  mechanical  connexion  or  attraction, 
it  is  desirable  to  have  it  designated  by  a  distinct  and 
peculiar  name ;  and  the  term  Affinity  has  been  employed 
for  that  purpose  by  most  modern  chemists.  The  word 
"  affinity"  in  common  language  means,  sometimes  resem 
blance,  and  sometimes  relationship  and  ties  of  family. 
It  is  from  the  latter  sense  that  the  metaphor  is  bor 
rowed  when  we  speak  of  "  chemical  affinity."  By  the 
employment  of  this  term  we  do  not  indicate  resem 
blance,  but  disposition  to  unite.  Using  the  word  in  a 
common  unscientific  manner,  we  might  say  that  chlo 
rine,  bromine,  and  iodine,  have  a  great  natural  affinity 
with  each  other,  for  there  are  considerable  resemblances 
and  analogies  among  them ;  but  these  bodies  have  very 
little  chemical  affinity  for  each  other.  The  use  of  the 
word  in  the  former  sense,  of  resemblance,  can  be  traced 
in  earlier  chemists;  but  it  does  not  appear  to  have 
acquired  its  peculiar  chemical  meaning  till  after  Boer- 
haave's  time.  Boerhaave,  however,  is  the  writer  in 
whom  we  first  find  a  due  apprehension  of  the  peculiar 
ity  and  importance  of  the  Idea  which  it  now  expresses. 


When  we  make  a  chemical  solution'"",  he  says,  not  only 
are  the  particles  of  the  dissolved  body  separated  from 
each  other,  but  they  are  closely  united  to  the  particles 
of  the  solvent.  When  aqua  regia  dissolves  gold,  do  you 
not  see,  he  says  to  his  hearers,  that  there  must  be  be 
tween  each  particle  of  the  solvent  and  of  the  metal,  a 
mutual  virtue  by  which  each  loves,  unites  with,  and 
holds  the  other  (amat,  unit,  rctinet]  ?  The  opinion  pre 
viously  prevalent  had  been  that  the  solvent  merely 
separates  the  parts  of  the  body  dissolved :  and  most 
philosophers  had  conceived  this  separation  as  performed 
by  mechanical  operations  of  the  particles,  resembling, 
for  instance,  the  operation  of  wedges  breaking  up  a 
block  of  timber.  But  Boerhaave  forcibly  and  earnestly 
points  out  the  insufficiency  of  the  conception.  This,  he 
says,  does  not  account  for  what  we  see.  We  have  not 
only  a  separation,  but  a  new  combination.  There  is  a 
force  by  which  the  particles  of  the  solvent  associate  to 
themselves  the  parts  dissolved,  not  a  force  by  which 
they  repel  and  dissever  them.  We  are  here  to  imagine 
not  mechanical  action,  not  violent  impulse,  not  antipathy, 
but  love,  at  least  if  love  be  the  desire  of  uniting.  (Xon 
igitur  hie  etiam  actiones  mechanicse,  non  propulsiones 
violenta?,  non  inimicitire  cogitanda?,  sed  amicitisc,  si  amor 
dicendus  copulas  cupido.)  The  novelty  of  this  view  is 
evidenced  by  the  mode  in  which  he  apologizes  for  intro 
ducing  it.  "  Fateor,  paradoxa  \\eec  assertio."  To  Boer 
haave,  therefore,  (especially  considering  his  great  influ 
ence  as  a  teacher  of  chemistry,)  we  may  assign  the 
merit  of  first  diffusing  a  proper  view  of  Chemical  Affinity 
as  a  peculiar  force,  the  origin  of  almost  all  chemical 
changes  and  operations. 

3.   To  Boerhaave  is  usually  assigned  also  the  credit 
of  introducing  the  norcl  "affinity"  among  chemists;  but 
*  Elemcnla  Chemice.     Lugd.  Bat.  1732,  p.  677- 


I  do  not  find  that  the  word  is  often  used  by  him  in  this 
sense;  perhaps  not  at  all*.  But  however  this  may  be, 
the  term  is,  on  many  accounts  well  worthy  to  be  pre 
served,  as  I  shall  endeavour  to  show.  Other  terms  were 
used  in  the  same  sense  during  the  early  part  of  the 
eighteenth  century.  Thus  when  Geoffrey,  in  1718,  laid 
before  the  Academy  of  Paris  his  Tables  of  Affinities, 
which  perhaps  did  more-  than  any  other  event  to  fix  the 
Idea  of  Affinity,  he  termed  them  "  Tables  of  the  Rela 
tions  of  Bodies ;"  "  Tables  des  Rapports :"  speaking 
however,  also,  of  their  "  disposition  to  unite,"  and  using 
other  phrases  of  the  same  import. 

The  term  attraction,  having  been  recommended  by 
Newton  as  a  fit  word  to  designate  the  force  which  pro 
duces  chemical  combination,  continued  in  great  favour 
in  England,  where  the  Newtonian  philosophy  was  looked 
upon  as  applicable  to  every  branch  of  science.  In 
France,  on  the  contrary,  where  Descartes  still  reigned 
triumphant,  "  attraction,"  the  watch-word  of  the  enemy, 
was  a  sound  never  uttered  but  with  dislike  and  suspi 
cion.  In  1718  (in  the  notice  of  Geoffrey's  Tables,)  the 
Secretary  of  the  Academy,  after  pointing  out  some  of 
the  peculiar  circumstances  of  chemical  combinations,  says, 
"Sympathies  and  attractions  would  suit  well  here,  if 

*  See  Dumas,  Leqons  de  Phil.  Chim.^  p.  3G4.  Rees'  Cyclopaedia^ 
Art.  Chemistry.  In  the  passage  of  Boerhaave  to  which  I  refer  above, 
affinitas  is  rather  opposed  to,  than  identified  with,  chemical  combina 
tion.  "When,  he  says,  the  parts  of  the  body  to  be  dissolved  are  disse 
vered  by  the  solvent,  why  do  they  remain  united  to  the  particles  of  the 
solvent,  and  why  do  not  rather  both  the  particles  of  the  solvent  and  of 
the  dissolved  body  collect  into  homogeneous  bodies  by  their  affinity  ? 
"denuo  so  affinitate  suae  naturae  colligant  in  corpora  homogcnea  ?"  And 
the  answer  is,  because  they  possess  another  force  which  counteracts 
this  affinity  of  homogeneous  particles,  and  makes  compounds  of  dif 
ferent  elements.  Affinity,  in  chemistry,  now  means  the  tendency  of 
different  kinds  of  matter  to  unite :  but  it  appears,  as  I  have  said,  to 
have  acquired  this  sense  since  Boerhaave's  time. 


there  were  such  things."  "Les  sympathies,  les  attrac 
tions  conviendroient  bien  ici,  si  elles  etaient  quelque 
chose."  And  at  a  later  period,  in  1731,  having  to  write 
the  clone  of  Geoffroy  after  his  death,  he  says,  "He  gave, 
in  1718,  a  singular  system,  and  a  Table  of  Affinities,  or 
Relations  of  the  different  substances  in  chemistry.  These 
affinities  gave  uneasiness  to  some  persons,  who  feared 
that  they  were  attractions  in  disguise,  and  all  the  more 
dangerous  in  consequence  of  the  seductive  forms  which 
clever  people  have  contrived  to  give  them.  It  was  found 
in  the  sequel  that  this  scruple  might  be  got  over." 

This  is  the  earliest  published  instance,  so  far  as  I  am 
aware,  in  which  the  word  "affinity"  is  distinctly  used 
for  the  cause  of  chemical  composition ;  and  taking  into 
account  the  circumstances,  the  word  appears  to  have 
been  adopted  in  France  in  order  to  avoid  the  word 
attraction,  which  had  the  taint  of  Newtonianism.  Ac 
cordingly  we  find  the  word  affinite  employed  in  the 
works  of  French  chemists  from  this  time.  Thus,  in  the 
Transactions  of  the  French  Academy  for  1746,  in  a 
paper  of  Macquer's  upon  Arsenic,  he  says "%  "  On  peut 
facilement  rendre  raison  de  ces  phenomenes  par  le  moyen 
des  affinites  que  les  differens  substances  qui  entrant 
dans  ces  combinaisons,  out  les  uns  avec  les  autres :"  and 
he  proceeds  to  explain  the  facts  by  reference  to  Geof- 
froy's  Table.  And  in  Macquer's  Elements  of  Chemistry, 
which  appeared  a  few  years  later,  the  "  affinity  of  com 
position"  is  treated  of  as  a  leading  part  of  the  subject, 
much  in  the  same  way  as  has  been  practised  in  such 
books  up  to  the  present  time.  From  this  period,  the 
word  appears  to  have  become  familiar  to  all  European 
chemists  in  the  sense  of  which  we  are  now  speaking. 
Thus,  in  the  year  1758,  the  Academy  of  Sciences  at 
Rouen  offered  a  prize  for  the  best  dissertation  on  Affinity. 
*  A.  P.  1740,  p.  201. 


The  prize  was  shared  between  M.  Limbourg  of  Theux, 
near  Liege,  and  M.  Le  Sage  of  Geneva*.  About  the 
same  time  other  persons  (Manherrf,  Nicolai  J,  and  others) 
wrote  on  the  same  subject,  employing  the  same  name. 

Nevertheless,  in  1775,  the  Swedish  chemist  Bergman, 
pursuing  still  further  this  subject  of  Chemical  Affinities, 
and  the  expression  of  them  by  means  of  Tables,  returned 
again  to  the  old  Newtonian  term;  and  designated  the 
disposition  of  a  body  to  combine  with  one  rather  than 
another  of  two  others  as  elective  attraction.  And  as  his 
work  on  Elective  Attractions  had  great  circulation  and 
great  influence,  this  phrase  has  obtained  a  footing  by  the 
side  of  Affinity,  and  both  one  and  the  other  are  now  in 
common  use  among  chemists. 

4.  I  have  said  above  that  the  term  Affinity  is  worthy 
of  being  retained  as  a  technical  term.  If  we  use  the 
word  attraction  in  this  case,  we  identify  or  compare 
chemical  with  mechanical  attraction ;  from  which  iden 
tification  and  comparison,  as  I  have  already  remarked, 
no  one  has  yet  been  able  to  extract  the  means  of  ex 
pressing  any  single  scientific  truth.  If  such  an  identi